Riva Kastoryano

Turkey/Europe: Space-Border-Identity

 But can Turkey's accession be assimilated to the enlargement process? To the extent that Turkey has been advised to respect the same political, economic, and social norms as the member states and to adopt the acquis communautaire, the law ensuring the functioning of the EU, its accession in principle resembles enlargement. But at the same time the nature of the debates and arguments produced by Turkey's accession make it a candidacy in a class of its own. With Turkey in Europe it is in fact a question of civilizational, cultural, and religious difference, and also of geography and borders: how far does Europe go?

One of the specificities of Turkey's candidacy is that is has served as a "mirror" for Europe -for its identity, its projects, its essence. It has brought out the ambiguities of its definition: a market, a political space, a civilization. It has underscored the paradoxes of its expectations and values: universality formulated in the terms of human rights and a -religious -European particularism which is now added to national particularisms. Turkey at the gate of Europe has thus made explicit a tension between Europe's "heritage," as expressed in the preamble to the Constitution, and its future, an unprecedented, ongoing political construction. 2 From this perspective, Turkey has been the alterity that has transformed a rational political project into an irrational discourse in search of a "collective consciousness" to define European belonging as an idea of unity in diversity.

The rejection of the Constitution by France and the Netherlands has not brought an answer to the question of what Europe is; to the contrary, it has led to a crisis that places the project itself in question, at once as market, as supranational political community, and as space. The opponents of the Constitution hung onto "national gains" they feel are threated; they expressed their fear of losing control of Europe's territorial and institutional limits as well as its diversity. But how are we to imagine the limits of diversity? Within this European space, how could we ignore the contributions of different national and/or minority cultures that make claims as nations, forming a common European culture and the modes

of expression of all collective identities, however complex and heterogeneous they may be? To this must be added the "non-Europeans" who arose from the immigration of the 1960s (including Turkish immigrants), Muslims established in various member states who seek to promote their so-called collective identities of "origin," be they religious, national, or ethnic, and seek new benchmarks in the European space.

For Turkey, Europe has been a choice of political modernization -a source of pride for the whole nation. The desire to be part of the European Union, to have a place in the "family photo," could thus be perceived as the confirmation of its choice to westernize at the end of the nineteenth century. Obviously, this is not the image that is returned to it by a united Europe, which, to the contrary, each time reminds it especially of its religious difference, even though by appropriating Western modernity in different historical circumstances and ways the Turkish nation-state had opted for constitutional secularism.

For Turkey, Europe is today an idea, a project, and an inspiration that serves to gear up its political, economic, social, and even cultural life. Social relations, national values, and political culture are evolving under the normative gaze of the European Union and its institutions. Civil society asserts itself through the active effervescence of associations in all domains and has a growing importance in political life. [START_REF] Murat | Between Turkey and Europe: Why Friendship is Welcome[END_REF] The criteria imposed by the Copenhagen Treaty, especially when it comes to human rights, minority rights, and gender relations, find an echo both in civil society and in the political class. All this is leading Turkey along the path of democratization and Europeanization.

But Europe is also a reality. It is a space of reference for the values, action, and mobilization of more than four million Turkish nationals settled in the different EU countries. Even if a large part of Turkish immigration transmits a negative image of Turkey and this image is used by opponents of its membership as an argument to reject it, another part circulating between the European space and Turkey mobilizes for equal rights both in the country of residence and in Turkey. While debates about Turkey place the identity, territory, and limits of the EU in question, this article seeks to show that, for activists in immigrant associations, Europe has already been drawn as a borderless social, cultural, and political space which, through transnational solidarity networks, includes Turkey in Europe. Their mobilization and their claims, directed at once at the countries of residence and at European institutions that act on Turkey, are pushing the latter to behave as a transnational actor subject to the rules of globalization, and, to a lesser extent, of Europeanization. [START_REF] Kastoryano | Les Turcs d'Europe et leur désir d'Europe[END_REF] The equation Turkey/Europe is thus inscribed within the problematic of Europe as a space of reference and action that operates through the integration of Turkish nationals as minorities in national societies and in a supranational political community. Such transnational actors serve as an engine of change and who, as in every interaction, reflect both to Europe and to Turkey their own contradictions in their reciprocal expectations.

A study published in December 2002 by the Center for Research on Turkey in Essen [START_REF] Ayça | Kiliçli for giving me all the recent studies of the Center for Turkey Studies at Essen on Turkish immigration in Germany and the integration of Turkey into the EU[END_REF] concluded by underlining that "Turks living in the member states of the European Union are responsible for building an important bridge between Turkey and the Union in the civil, economic, and political domains." 6 This political and economic role had already been attributed to them by both the media and the political class with the creation of a new social category: Turks abroad. This appellation describes Turkey's desire to see part of its population mobilize for its accession to the EU. But for the populations concerned the same appellation implies a solidarity that transcends the borders of the Union, structured by formal or informal transnational networks. The activists who are building these networks aim, by means of European representation, to make themselves heard in the receiving country and indirectly in their country of origin, Turkey. This de facto includes Turkey in the claims for equal rights and justice especially in minority issues or religious rights. The role of economic, cultural, and/or political intermediary attributed to Europe's Turkish nationals or migrants from Turkey in this context is expressed by transfers of identities, in most cases are redefined in relation to receiving countries, toward the country of origin. This implies reciprocal influence and interdependence. Mobilization at the European level through solidarity networks is related to Turkey's European integration, which itself bears on modalities of integration in the European countries -whence the interdependence that makes Turkey part of Europe.

This activism goes back to the 1980s, when migrants from Turkey -like other immigrant populations in Europe -developed a discourse around the permanence of their presence by affirming that they were "here to stay." For their part, the receiving countries set up policies to assure their integration, notably by providing them with resources to create associations as centers of solidarity and intermediaries with the public authorities in each country. New political actors emerged and reorganized their interests and social, cultural, ethnic, or political identities within the framework of associations created in most cases with the support of the countries of residence in the name of a democracy now concerned to recognize differences. Some of these associations replaced the militant, revolutionary, religious, or ethnic organizations of the right or the left, rooted in Turkey and transposed in "exile," that were principally oriented toward Turkey; they took the Turkish state as their adversary and protested against Turkish policy, its conception of the nation, its conception of religion, and its practice of democracy. These associations, financed by individuals, (opposition) political parties, or international organizations, found a field of action and an expression of identity in various European countries that they had never known in Turkey. Other associations were created, often with the help of the public authorities of the receiving countries, to promote the integration of immigrant populations, including those from Turkey. Turkish public authorities value their activities, and their representatives are regarded as ideal interlocutors between the two countries. Still others have tried to combine these two aspects (being against official Turkey and for integration in the receiving country) in order to gain credibility with the Turkish immigrant population as a whole. But, on the whole, whether the associations derive from political groups already active in Turkey, from the conversion of Turkish or European workers' movements, or from local initiatives in the receiving countries, the discourse of their leaders gives a privilged place to culture and identity. The associations thus reflect all the diversity of complex societies: social (class, sex, age), cultural, linguistic (Turk/Kurd), and religious (Sunni/Alevi). It is as if all the ethnic, cultural, linguistic, and religious differences that had been repressed by the Turkish nation-state's concern for cultural and linguistic homogenenization reemerged in one form or another, "liberated" in the country of immigration, where each particular trait is an element of distinction. To these divisions can be added the ideological divisions that affect some Islamic religious associations, the stand they take for or against the powers that be in Turkey as well as the policies of receiving countries.

These associations are included within the existing institutional structures of the receiving countries, and activists in these associations have adopted the political "rules of the game" of these countries in order to position themselves vis-àvis the state and negotiate their claims on equal terms. The so-called politics of identity and recognition in the European countries found an echo in the actions and discourse of association leaders. Their definition of identity varied depending on their ideology: national identity, religious identity, ethnic identity, political identity…Their claims bear on equal social, cultural, and political rights, the fight against discrimination and racism -in short, citizen rights as well as the recognition and representation of cultural and religious specificities.

With the European contruction, the quest to have identities represented went beyond the framework of the state and now extended to the European level. Just like claims for residency, citizenship rights, or protection against deportation now addressed to European institutions, interests expressed in terms of the identities of populations of immigrant origin, like states themselves, find a field of action in a Europe in construction, leading to new forms and structures of representation and new negotiations.

Whence the elaboration of new strategies that go beyond national frameworks and situate immigrant populations beyond their relations to their host countries. This allows them to strengthen their claims on both the national and the European level; transnational networks are woven and new structures of solidarity are drawn across across European borders, stimulating associational networks to join the "spider's web" of professional networks that cover already the European space. The aspirations and activities of Turkish associations were connected to a European Parliament project, which in 1986 provided so-called immigrant associations with funds to coordinate their activities. Out of this initiative a new transnational structure was born: the "Migrants' Forum," dissolved in 2001. While the Forum owed its creation to Union budgetary policy, it nonetheless sought to become "a place of expression for non-Community populations established in Europe, through which they can make their claims but also diffuse information from the European authorities." [START_REF] Neveu | Citoyenneté ou racisme en Europe: exception et complémentarité britanniques[END_REF] The associations, whose activities were supported by the welfare states of the member countries and whose activists elaborated a discourse of human rights and their universality, saw transnational mobilization as an effective way of struggling against racism and xenophobia. 8 The Forum was not very significant in itself. But it had the merit of presenting Europe to populations of immigrant origin as a new transnational political space where solidarity networks compete to circumvent states by acting directly on supranational European institutions. This allowed them to strengthen their action on both the national and the European level, eliciting identification among immigrant populations with a political ensemble that transcends national frontiers. This is due to the very nature of the EU, where the logic of supranationality created a transnational civil society in which (national, regional, religious, and professional) compete and collaborate, manifesting all the fragmentation of democratic societies. The politicization of each network is the root of the emergence of a transnational space. Owing to the density of interactions among different actors from different national traditions, this creates a new space of political socialization for the individuals and groups that create it. It becomes a space in which these actors learn a new political culture that takes shape beyond national frameworks and institutions. For Turkish migrants, this new stage of political socialization was added to others -for some, Turkey before immigration; for others, the associational framework of the receiving countries. This dynamic, born of political participation in several spaces at once and accelerating the interactions among different value institutions and political cultures, constitutes the basis of transnational action. [START_REF]and Blanc-Szanton Nations Unbound: Transnational Projects, Postcolonial Predicaments and Deterritorialized Nation-states[END_REF] The transnational activism of Turks in Europe includes the populations that have resulted from Turkish immigration into various European countries and Turkey itself, making the latter a source of ethnicity and Europe a source of legitimacy for mobilization and claim-making. But the identity constitutive of their action does not necessarily correspond to the "official" identity proclaimed by Turkey. Indeed, the demand for recognition makes all the differences within the national group reappear in the public space. These differences are not only expressed in relation to France or Germany; they also and above all emerge within representations of Turkey. Each particular trait now constitutes an element of distinction redefined through a process of "reappropriating" identity suggested by identity politics of European countries -Kurds, Turks, Alevis, Sunnis -that transcends the territorial borders between Turkey and Europe. Their goal is to increase the influence of these identities in immigration and thereby to act more effectively in Turkish political life by sensitizing international opinion and supranational institutions to their cause. This influences juridical practices in Turkey, especially concerning the Kurdish minority, and is expressed by the constantly rising number of appeals against Turkey to the European Court of Human Rights. [START_REF] Bockel | La Turquie et l'Europe: Une coopération tumultueuse[END_REF] Turkish migrants thus circulate between these spaces, carrying the values and political norms acquired in immigration, institutionalized within the framework of associations, and legitimized by national and European authorities. This is the case of the Kurds, for example; through the voice of their activists, Kurdish immigrants seek recognition as a "Kurdish community" with a culture, history, and language distinct from those of the "Turkish community," mainly in Germany. This differentiation, which takes the form of a conflict of "nationalisms" in Turkey, puts the Kurds in a "minority" position within a transnational Turkish community. Their demand for recognition brings forth claims elaborated not only as immigrants but based on the status of a double minority: a minority within a minority in Germany and a minority in Turkey. So it is that, possessing more political resources in Europe than in Turkey, they bring their claims to the European Court of Human Rights, the Council of Europe, the European Court of Justice, and the European Parliament in order to return to Turkey to obtain recognition as a "miniority" from Ankara and secure respect for their cultural and linguistic specificities in public institutions. It is thus now by means of European institutions that populations defined as "minorities" or "communities" in a national (and European) context obtain both recognition and the ability to negotiate with the country in which they live as well as in Turkey.

The concept of a minority is not, however, without ambiguity. It is the source of uncertainties in establishing juridical forms of recognition. The definition proposed by the European Convention on Human Rights is broad: "the term minority designates a group numerically smaller than the rest of the population whose members seek to preserve their culture, traditions, religion, or language." In Turkey, Kurds assimilated to the Turkish nation-state by virtue of belonging to Islam do not enjoy institutional representation of a specific cultural identity. The combination of cultural, linguistic, and territorial belonging that defines the Kurds as a minority in Turkey is accompanied by a politics of recognition that today is officially imposed on Turkey by the Copenhagen criteria.

Another, more controversial identitarian element is Islam, the "national faith," [START_REF] Halman | Islam in Turkey[END_REF] and its political use in Europe, which is not unconnected to the evolution of Islam in Turkey. Indeed, the Islamist association Millî Görüs (National Vision), an organ of the prosperity party in the Refah period, was initially founded in Germany, later becoming the Fazilet (Virtue) Party -the ancestors of today's governing Justice and Development Party (AKP). The raison d'être of organization in immigration was associated with cultural associations in order to assist affected populations by responding to their cultural and religious needs. In 1990 Secretary General Ali Yüksel, a 45 year-old engineer and theology graduate described by the media as "modern," was declared Seyhulislam, religious head of all Muslims (a status abolished with the Ottoman Empire). By creating the seat of the "European Organization of the National Vision" in Cologne, he declared this Rhineland city the node of the Turkish Islamic network in Europe. [START_REF] Sullivan | [END_REF] Since the 1980s, this organization, banned in Turkey until the AKP won power, was not only the largest in the immigrant community but also the best established in Europe, with 28 branches. Was it not this political know-how, acquired in a democratic space (Germany) and transposed into Turkish political life and more specifically into power, that has made a place for the expression "moderate Islam," which the AKP represents in the eyes of international and Turkish opinion? Should we then be surprised that the AKP is assimilated to a Christian Democratic party?

For its part, in 1981 the Turkish state, at war against political Islam in Turkey and in the immigrant community, established the Directorate of Religious Affairs (Diyanet), connected to the Prime Minister through a network association called the DITIB in all European countries. Its objective is to fight political Islam as it has developed among immigrants influenced by Millî Görüs and fraternal associations and the "dissident" Islam spread by political parties that oppose the secular principles of the Republic. The secular state thus explicitly introduces and institutionalizes religion into its principles as an element of national identity.

As for fraternal associations (tarikat), freed in immigration from the "republican prohibition," they reappear in Turkey as civil society associations and correspond to interest groups that participate in politics. Their role is incontestable in restoring the relations of forces within national political life and defining the place of Islam as popular culture and as belief to be taken into account while seeking an equilibrium. [START_REF]This is at least the message of Fetullah Gülen, head of the Nurcu movement. For more on the brotherhood, see Serif Mardin, Religion and Social Change in Turkey: The Case of Beddiuzzaman Said Nursi[END_REF] On the other had, the Alevis, who had been assimilated to the tarikats, their system of belief maintained in secret, even taboo, and consequently banned from the public space, have benefited from increased visibility since the 1990s. 14 The question is to what extent their current visibility and integration into Turkish political life is connected to their image in immigration. In fact, the Alevis attracted the attention of public authorities in European cities by their "distance" from religion and radical Islamic practices. To what extent has their representation of an undogmatic Islam, even a modern Islam (some question whether it is Islam at all), in Turkey as well as in immigration, played a role in changing the attitude of the political class in Turkey, which has all but recognized them officially? Or is it because the Alevis affirm the principles of a secular state, so that recognizing them is a way of combating ethnic and religious divisions?

We are thus seeing an inversion that implies a redefinition of relations between spaces and political actions. At the beginning of immigration, populations from Turkey transposed the cleavages and political and ideological conflicts of social classes to the European countries, making Europe an extention of Turkish political life. Today, the mobilization in Turkey of organizations enjoying recognition and legitimacy in immigration expresses a political transfer that gives new élan to mobilizations and claims in Turkey and leaves neither public opinion nor the political class indifferent. These transfers operate in the cultural, economic, and social domain and act in both directions. They are amplified by new means of communication, especially satellite broadcasting, which burst upon Kurdish or Turkish, Sunni or Alevi families from Turkey. Just as Turkish immigrant families in Europe experience Turkey everyday via images transmitted by 12 or 14 private and public channels, Turks in Turkey watch European channels. [START_REF]L'immigration turque et kurde: La dynamique ségmentaire, la nouvelle donne générationnelle, et le nouvel ordre communicationnel[END_REF] These evolutions blur the connection between territory and identity: the territory of belonging remains regional, the territory of reference becomes national or religious (still to be negotiated), and the territory of residence is French, German, Dutch, or simply European.

These transfers integrate both the Turkish state and Turkish civil society associations into Europe as a political, social, or economic actor. Indeed, the Turkish state has acted on Turkish immigration from the beginning through language instruction. Its intervention is included within the framework of bilateral agreements between different receiving countries. The constant presence of the Turkish state in immigration helps maintain immigrants' attachment to their country of origin as if to maintain the idea of Turkish citizenship as extraterritorial belonging, beyond the national territory. From this perspective Turkey dovetails with one the characteristics of the European space, which de facto and de jure enourages extraterritorial citizenship by granting political rights (local and European elections) to citizens of the Union (Maastricht Treaty, Art. 8). This deterritorializes, even denationalizes, the European space. Obviously the Turkish state opposes this process to the extent that it seeks to reclaim its citizens at a distance, but all the same it behaves as an actor that is integrated into the process of Europeanization.

The Turkish state intervenes also in the mobilization of nationalist forces beyond its national territory, opposing movements that oppose its national interests and nation-state principles. By helping official Turkish organizations reconstitute themselves as associations in Europe, the Turkish state contributes to the development of state nationalism just like the nationalist movements that oppose it, like the Kurdish movement or Turkish Islamic movements. Many Kemalist associations are active in Turkey, like the Association of the Thought of Ataturk, amplifying their mobilization and diversifying their activities in Germany and affirming their presence with and against other associations. This shows that the Turkish state participates at the same time in the associative movement and their claims to "difference" which are integrated into the framework of legitimacy of European states.

In short, the politicization of identities reshapes relations between migrants from Turkey and receiving countries. Turkey, driven by a concern with democracy and human rights, cooperates with some associations, participating in the multicultural policies of European countries. In addition, through its consular network it cultivates relations with social and cultural associations created by migrants, opening relations with them that have a local and national impact on families. This does not contradict the claims of the Kurds, whose nationalist movements developed in Europe and then found an echo in Turkey, nor those of religious associations, even if it sometimes complicates diplomatic relations between European countries and Turkey. They act as "guarantors" of the Copenhagen criteria, and consequently of the path to European accession.

The objective of the Turkish state is clear: it seeks to present an image of Turks abroad, united around Turkey as a national community, as an ideal by situating itself in relation to different aspects of Turkish identity -national, ethnic, religious, political. The implicit objective is to oppose political forces that act against Turkey and openly oppose Turkey's accession to Europe, always in the name of human rights, minority rights, or religious freedom, with demonstrations at the European Commission or the European Parliament. Political actors who come out of Turkish immigration in Europe respond to the expectations of the Turkish state by trying to form lobbies recognized in both countries, based on Community institutions, determined not only by Turkish political life but situating themselves in the German system or opposing it. They also react to all statements by the German, French, or Austrian government concerning Turkey and to Ankara's policies when they thwart EU expectations.

Their power is based on the economic success of some. Especially since the fall of the Berlin Wall, transnational economic actors, increasingly organized in many regions into Turkish businessmen's associations in Germany, have sought influence in the national politics both of their country of residence and Turkey. 16 Putting economics at the service of politics, businessmen of Turkish origin, who are more and more numerous in Europe, have the power to negotiate with the authorities in both spaces.

These organizations are supported by the associations of Turkish civil society, especially TUSIAD, which is very influential in national political life as well as in Turkey-US and Turkey-EU relations. The objective of this association of Turkish industrialists and businessmen, founded in 1971, is "to promote public wellbeing through private enterprise"; it is most active in national politics, but extended its field of action with an office in Brussels in 1996 and in Washington in 1998. In September 2003 the association opened its Berlin office. In his speech at the opening, the president of the association's administrative council explained the choice of Berlin:

Berlin is a historically important city and will be more so in the future. It is the capital of our largest economic partner, Germany. It is the center of German citizens of Turkish origin, whose number is continually growing. It is becoming the premier economic and political center in Europe. By virture of its dynamism and creativity, combined with its history, this city is now at the same time a world capital.

Thus, actions and political strategies emanating from Turkey (state and civil society) are conjugated with those of Turkish immigrants established in Europe. Together they contribute to the integration of Turkey into Europe. Political experience, the democratic know-how acquired in Turkish immigrants' struggle for equal rights and legal citizenship, are appropriated by Turkey in its strategies of Europeanization.

The presence of a Turkish immigrant population in Europe and its transnational mobilization underlines the role of minorities in international relations and their influence in European national as well as international opinion. The issue is not new. Many studies, especially in the United States, have shown the role of migrants in international relations. [START_REF] On | Migrants, Refugees and Foreign Policy: US and German Policies Toward the Countries of Origin[END_REF] In Europe the intervention of supranational institutions in creating networks and diffusing norms vis-à-vis both states and organizations that act directly at the European level orients actions. [START_REF]Transnational Social Movements and Global Politics: Solidarity beyond the State[END_REF] The mobilization of immigrant activists is thus at once a challenge and an opportunity for Turkey: a challenge with respect to its ability to respond to their expectations, all the more so since these expectations dovetail with those imposed by European normative supranationalism; an opportunity by taking Turkey into a process of globalization that assures its inclusion in a process of Europeanization.

The effects of this mobilization also underline the political importance of Turks abroad in redefining the representation of the Turkish nation, its values, its culture, and its official identity. The claims for recognition of the Kurdish identity in particular have put the Kemalist-based official discourse, a nationalism considered "natural" until recently, into an increasingly defensive position among immigrant populations and in Turkey. Similarly, Islam as a culture and an identity that is, like other religions in various European countries, finding a place within the framework of public institutions is today leading Turkey to redefine its secularism and to seek other models than the French, which is also on the defensive. [START_REF] Karslsson | Turkey's Cultural and Religious Heritage -An Asset to the EU[END_REF] Here we see one of the paradoxical effects of Europe, which, despite the talk of rejection as the reality approaches, has considered Turkey an "eligible" country from the beginning thanks to this exemplary combination of secularism and Islam, which is now leading Turkey to reexamine precisely what makes for its exceptionality in the Muslim world. From this perspective, European integration is a challenge for Turkey in its quest for a new equilibrium of politics and religion.

Turkey's membership is a challenge for Europe. Europe will need to define its identity -another way of identifying itself than "othering Turkey," to take up Nilüfer Göle's expression, [START_REF] Göle | Interpénétrations. L'Islam et l'Europe[END_REF] and if need be defining its borders. On a pessimistic note, Philippe Reynaud thinks that "if immigrants become Europeans it is because, having had to disavow themselves, they are integrated into a historical community that has been constituted through the tentions between Christianity and modernity"; he adds that "today it is hard to see how a community with no identity aside from human rights could be a stabilizing factor in a world where the decline of classic wars has by no means made tensions disappear." [START_REF] Ph | L'Europe peut-elle avoir des frontières?[END_REF] Can we not see Turkey's accession as a contribution to stability and the reduction of the tensions and conflicts Huntington called conflicts of civilizations? As for the definition of borders, at this stage nothing can prevent Europe from being a transnational space, but we could not think that "re-territorialization" would consolidate Europe as a supranational entity.

Expert reports already emphasize a change in the course of the European construction. The rejections of the Constitution have plunged Europe into a crisis and decision-makers into deep reflection on anticipated changes. The prospect of Turkish membership can only accelerate new dynamics, which should be considered for their functioning and acceptance by the "peoples of Europe."

It is here that the true challenge still lies for Europe: accepting Turkey as part of the "peoples of Europe." At the time of the opening of TUSIAD in Berlin, its president expressed his wish to "see Europe succeed in maintaining its multiculturality in the face of the test of Turkey." Referring above all to Turkish immigration in Germany, he added that "using the gains of Turkish-Germans or German-Turks and their economic and cultural contributions to Germany, Germany and the other European countries will have to have a rational approach to the question of Turkey's accession."

But European multiculturalism is beyond relations between immigrants and states; it is located in the relations and reciprocal recognition of member states. [START_REF] Kastoryano | Quelle identité pour l'Europe? Le multiculturalisme à l'épreuve[END_REF] The question of European multiculturalism always raises the question of combining Europe's universalistic ideology and the cultural-historical particularism that characterizes each of its member states. The European political project cannot ignore the factual plurality within which different national cultures already express and impose themselves. But the irrationality of the debates around Turkey's accession threatens to lead to the transformation of the European "project" into a particularism that could easily be qualified as a "Euro-nationalism" and would obliterate the bases of its creation and the values it has sought to embody and spread.

The debates and reports on relations between Turkey and Europe recall the path to be travelled on each side. [START_REF] Grabbe | From drift to strategy: The case of Turkey's accession[END_REF] For the Turkish side it is a matter of meeting the conditions to become part of European multiculturalism, from its plurality to its equality. On the side of the member states at stake is a true universal opening, and their effort to influence public opinion on this view of Europe. For this, it is necessary that good wills agree to embrace the idea that Turkey as member state is not a "threat" to European identity, but to the contrary a trump in its representation of universality.

Translated by James Ingram

NOTES


N Kanari
email: ndue.kanari@univ-lorraine.fr

N Menad

F Diot

E Allain

J Yvon

PHOSPHATE VALORIZATION BY DRY CHLORINATION ROUTE

Keywords: Dicalcium phosphate dihydrate, Calcium pyrophosphate, Chlorination, Valorization

This work deals with the extraction of phosphorus chlorinated compounds from phosphate materials using chlorination with gaseous chlorine. An industrial sample of dicalcium phosphate dihydrate, after transformation into calcium pyrophosphate (Ca 2 P 2 O 7 ), is subjected to reactions with Cl 2 +CO+N 2 and Cl 2 +C+N 2 at temperatures ranging from 625 to 950 °C using boat experiments. Gathering results of the thermodynamic predictions and TG/DT analysis with those of SEM and XRD examinations of the chlorinated residues allowed the interpretation of phenomena and reactions mechanism occurring during the calcium pyrophosphate carbochlorination. Reaction rate of Ca 2 P 2 O 7 by Cl 2 +CO+N 2 at 950 °C is slowed down due to the formation of a CaCl 2 liquid layer acting as a barrier for the diffusion of the reactive gases and further reaction progress. While, the carbochlorination with Cl 2 +C+N 2 led to almost full chlorination of Ca 2 P 2 O 7 at 750 °C and the process proceeds with an apparent activation energy of about 104 kJ/mol between 625 and 750 °C. Carbochlorination technique can be considered as an alternative and selective route for the valorization of low grade phosphates and for the phosphorus extraction from its bearing materials.

Introduction

Phosphorus, based in a vast variety of the phosphorus minerals, especially phosphates, is vital for flora, fauna and human life. Most of phosphate bearing materials are used in the agriculture as fertilizer and another important end use of phosphates is detergency. Further, it seems that there are no consistent substitutes for phosphorus in agriculture. According to available statistic data [START_REF] Survey | Phosphate rock statistics[END_REF], the world production of phosphate rock (26-34 % P 2 O 5 ) has more than tripled during the last fifty years reaching 217 million tons by 2012. With this increase production rate and by taking into account that the phosphate natural resources are limited, the depletion of the economically viable phosphorus reserves is evident within a near future. One way to overcome this difficulty should be the exploitation and upgrading of low grade phosphate ores and recycling of the phosphorus constituents contained in the wasted materials. A coherent overview of the techniques (physical and thermal routes) used for producing high-grade phosphate products suitable for fertilizers and other phosphate compounds is given elsewhere [2].

Thermal route is also used for the reduction of phosphates by carbon in presence of SiO 2 at temperatures, often, as high as 1600 °C. The elemental phosphorus is produced followed by its subsequent oxidation to phosphorus pentoxide which is an important precursor for the H 3 PO 4 synthesis. The research works related to the action of gaseous chlorine on the phosphates are not found in the literature or they are based on old studies. Recent few works are focused on the behaviors of phosphorus compounds during removal of heavy metals contained in the sewage sludge ash using thermochemical methods, especially, chlorination [3][4][5]. However, a good number of investigations are devoted to the chlorination technique for the separation and extraction of valuable metals from raw and residual materials [6][7][8][9][10][11][12][13]. Other reports concerned the reactivity and kinetics of interaction of chlorine with several rare earth elements oxides such as CeO 2 [14], Sm 2 O 3 [15], Y 2 O 3 [16], Nd 2 O 3 [17] and La 2 O 3 [18].

In this context, this study is devoted to the possibility of the phosphate valorization and extraction of phosphorus chlorinated compounds by dry chlorination route. As the first step of the investigation, the results presented here are almost related to the reaction of dicalcium phosphate dihydrate (CaHPO 4

. 2H 2 O) with Cl 2 +CO+N 2 and Cl 2 +C+N 2 under isothermal conditions. Both thermodynamic and kinetics aspects of the carbochlorination reactions are discussed in order to achieve a selective separation process of the phosphate constituents. This work is an extension of our most important findings concerning the chlorination processes summarized earlier [19], as well as of some kinetics of the chlorination reactions of various oxides in presence of reducing and /or oxidizing atmosphere [20][21][22][23][24].

Materials and experimental procedures

A wet sample of dicalcium phosphate dihydrate (CaHPO 4

. 2H 2 O), known also as brushite, of industrial origin, is used for this study. CaHPO 4 . 2H 2 O represents about 98.0 % of dried solid of the sample. Raw wet sample is heated at 150 °C and X-ray diffraction (XRD) analysis revealed the presence of CaHPO 4

. 2H 2 O and neo-formed monetite (CaHPO 4 ) as main crystallized phases. This sample is mostly used for the carbochlorination tests and it is designated as DCPI. Both samples (wet and dried samples) are subjected to the thermogravimetric (TG) analysis under N 2 atmosphere over 20-1000 °C temperature range with a linear heating rate of 5 °C/min. Results are depicted in Fig. 1 as evolution of the percent mass loss (% ML) as a function of the temperature. Solid products obtained at 200 °C, 600°C and 1000°C are examined by XRD. The % ML observed for the wet sample at temperatures lower than 100 °C corresponds to the free water removing of the wet brushite sample. Transformation of brushite into monetite occurred between 100 °C and 200 °C, as confirmed by XRD. The next step of % ML observed at T > 400 °C is attributed to the thermal conversion of the monetite into calcium pyrophosphate (Ca 2 P 2 O 7 ) which is stable up to at least 1000 °C.

Charcoal is chosen as carbon bearing substance for the carbochlorination of DCPI. XRD analysis showed two broad diffraction peaks attributed to slightly graphitized carbon structure. Used gases (N 2 , CO and Cl 2 ) are of a high purity (99.9 wt %). Boat chlorination experimental tests were performed in a horizontal setup described previously [22]. This setup is composed of a gas measuring unit followed by a gas purification item and a horizontal electric furnace. The chlorination residues were examined by scanning electron microscopy-energy dispersive spectroscopy (SEM-EDS) and XRD.

Results and discussion

Thermodynamic elements

As noted in the previous section, thermal treatment of CaHPO 4

. 2H 2 O and CaHPO 4 under nitrogen atmosphere at temperatures higher than 500°C led to the formation of the calcium pyrophosphate (Ca 2 P 2 O 5 ). Therefore, Ca 2 P 2 O 5 is considered as only compound for the thermodynamic calculations of chlorination (Cl 2 ) and carbochlorination (Cl 2 +CO; Cl 2 +C) reactions. The evolution of free standard energy changes (G°) as a function of the temperature for the overall reactions of Ca 2 P 2 O 7 with Cl 2 , Cl 2 +CO and Cl 2 +C (Eq.1 through 9) are calculated by using thermochemical databases [START_REF] Barin | Thermochemical data for pure substances, Parts I and II[END_REF][START_REF] Roine | Outokumpu HSC Chemistry® for Windows, version 3.0[END_REF] and the obtained results are traced in Fig. 2.

(

(2)

(3) (4) (5) (6) (7) (8) (9)
Chlorination of Ca 2 P 2 O 5 by chlorine in absence of reducing atmosphere seems to be not feasible since the value of G° are higher than 100 kJ/mol Cl 2 in whole temperature interval studied and whichever is the phosphorus chlorinated species (Fig. 2a).

+ + = + + CO 2
Carbochlorination reactions of the calcium pyrophosphate with Cl 2 +CO and Cl 2 +C (Fig. 2b and2c, respectively) are characterized by negative values of the (G°) resulting to a favorable process from thermodynamic point of view.

The phase predominance area diagrams (Fig. 3) for the system Ca-O-Cl and P-O-Cl as a function of temperature and partial pressure of chlorine at two fixed partial pressures of oxygen (pO 2 ) are established by using HSC Chemistry thermochemical database [START_REF] Roine | Outokumpu HSC Chemistry® for Windows, version 3.0[END_REF]. Calcium chloride is the predominant phase in Ca-O-Cl system at high partial pressure of chlorine at two chosen values of pO 2 (Fig. 3a and3b). As could be expected, its predominance area is extended at low oxygen potential (pO 2 = 1.013x10 -18 kPa). Phosphorus pentoxide (with molecular formula P 4 O 10 ) is the only stable phase in the P-O-Cl system for pO 2 = 10.13 kPa at whole temperature range studied (it is not shown in the Fig. 3). However, phosphorus oxychloride (POCl 3 ) and trichloride (PCl 3 ) are the most stable phases for temperatures higher than 300 °C and 600 °C, respectively for low oxygen potential (pO 2 = 1.013x 10 -18 kPa) and for high partial pressure of chlorine (Fig. 3c). Decreasing the oxygen partial pressure at values lower than (pO 2 = 1.013x10 -18 kPa) leads to the formation of phosphorus trichloride as the main phase in equilibrium with P 4 O 10 . In summary, these thermodynamic data indicate that Ca 2 P 2 O 7 can be chlorinated by chlorine only in presence of a reducing agent leading to the formation of the phosphorus chlorinated compounds.

Another interesting thermodynamic knowledge of the above mentioned systems is the vapor pressure of the synthetized chlorides. Fig. 4 gives the vapor pressure of phosphorus, iron and calcium chlorides as a function of reciprocal temperature. It is clear from this figure that the large difference between the vapor pressure of phosphorus chlorides (PCl 3 , POCl 3 , PCl 5 ) and that of CaCl 2 will allow their separation. Similarly, these phosphorus chlorides can also be separated from iron chloride by fractional distillation and/or by controlled cooling of the gas phase. Oxidation of these phosphorus chlorides leads to the synthesis of phosphorus pentoxide.

This thermodynamic study summarizes the basic elements for a selective extraction the phosphorus chlorinated compounds from calcium phosphate during its chlorination, while the kinetics parameters of the process will be revealed through the experimental results described in the following sections.

Carbochlorination with Cl 2 +CO+N 2

The first carbochlorination tests of the DCPI were performed at 950°C on about 3 grams of sample, when the reaction time was varied from 0.5 to 4.0 hours. A gaseous Cl 2 +CO+N 2 mixture with a total gas flow rate of 100 L/h containing 30 % Cl 2 +CO and having a Cl 2 /CO molar ratio equal to 1 is used for the experimental tests. This high gas flow rate aims at avoiding reactive gas starvation and minimizing the mass transfer phenomena. Results are represented in Fig. 5a as an evolution of the sample % ML versus reaction time. A rough examination of these data indicates that % ML increased almost linearly with the reaction time. Such % ML curve shapes recall a gassolid process with an overall rate governed by the reaction rate occurring at infinite slabs and/or the process controlled by the volatilization rate of liquid reaction products. SEM-EDS spectra of the DCPI initial sample and that of the carbochlorination residue obtained after 4 hours are showed in Fig. 5b. These spectra are almost identical except the presence of chlorine in the chlorination residue which is attributed to CaCl 2 . Note that all phosphorus chlorinated compounds are highly volatile at 950 °C (see Fig. 4). This result suggests that the presence of the liquid calcium chloride affects the chlorination process. It probably covers the DCPI particles acting thus as a barrier for the diffusion of the reactive (Cl 2 +CO) gas to the reaction zone. The progress of reaction then depends on the volatilization rate of liquid CaCl 2 .

N. Kanari et al. / JMM 52 (1) B (2016) 17 -24

To have an idea about the thermal behavior of the calcium chloride, a TG analysis coupled with differential thermal (DT) measurement is performed under nitrogen and the obtained results are shown in Fig. 6.

Data recorded above 750°C are the most important for this study. The endothermic peak at 784°C is attributed to the melting of CaCl 2 and it is close to the fusion point (775°C) of the calcium chloride found in the literature [START_REF] Roine | Outokumpu HSC Chemistry® for Windows, version 3.0[END_REF]. Then, the mass loss recorded beyond this temperature is due to the volatilization of the liquid calcium chloride. A comparison of these results with the previous ones, indicates that the carbochlorination rate of the calcium phosphate is affected by the physical properties, i.e. melting point and volatilization rate, of the synthetized calcium chloride. For these reasons it was suggested to achieve the chlorination of DCPI at temperatures lower than the melting point of CaCl 2 . Further, the solid carbon is used as reducing agent instead of gaseous carbon monoxide which can disperse the solid CaCl 2 product creating free paths for the chlorine diffusion.

Carbochlorination with Cl 2 +C+N 2

Mixtures of DCPI and carbon with various C/DCPI molar ratios were prepared. The most appropriate mixture for the carbochlorination tests is chosen to be (DCPI+C) having 1.5 times more carbon than the stoichiometric amount required according to Eq. (7). A gaseous Cl 2 +N 2 mixture with a total flow rate of 100 L/h containing 30 % Cl 2 is used for the chlorination. Experimental tests were carried out under isothermal conditions in the range 625 -900 °C for a reaction time of 8 h. Results are plotted in Fig. 7 as evolution of % ML of the DCPI+carbon sample vs temperature.

As seen in Fig. 7, a regular and almost exponential function shape of %ML of the sample vs temperature is recorded for the treatment temperature up to 750°C. Afterward, the reaction extent decreased and temperatures as high as 900°C are necessary to have the similar %ML as that obtained at 750°C. Again, the complications of the DCPI carbochlorination at high temperatures are interpreted assuming that the process is affected by fusion of CaCl 2 . Similar phenomena have been reported previously for the carbochlorination of MgO [22].

Attempts were made to follow the evolution of the elemental and mineralogical composition of the treatment residues at different carbochlorination temperatures. General spectra of the SEM-EDS N. Kanari analysis regarding to the initial sample (DCPI+carbon) and to the carbochlorination residues are grouped in Fig. 8.

As expected, the initial sample is composed of Ca, P, O and C. The carbochlorination residue at 650°C displays a broad peak of chlorine which can be assigned to calcium chloride. The spectra of the residue obtained at 700°C and especially of that produced from 750°C are characterized by a significant decrease of the phosphorus peak intensity indicating its remove from the residue due to high volatility of the phosphorus chlorides already generated during reaction of Cl 2 +CO with DCPI. The gas phase of the carbochlorination tests is cooled at room temperature and no substantial solid condensate is observed during treatment at T≤750°C. This observation reinforces the hypothesis that the most likely chlorinated compounds of phosphorus is PCl 3 having a sufficient vapor pressure at room temperature (see Fig. 4) to leave with exhausted gases.

The results of the XRD analysis applied to the raw sample (DCPI+carbon) and to the treatment residues are summarized in Table 1 Note that these hydrated species are the predominant phases of the carbochlorination residues at temperatures higher than 675 °C. The calcium pyrophosphate (Ca 2 P 2 O 7 ) phase is detected by XRD up to 725 °C, indicating that the almost full breakdown of the Ca 2 P 2 O 7 structure is achieved beyond this temperature. The presence of carbon in the carbochlorination residues is also detected during the treatment at each temperature between 625 and 775 °C. This is due to the fact that the quantity of carbon used is about 1.5 times higher than that of amount assumed for the carbochlorination reactions. Further, some isothermal tests of carbon in Cl 2 +N 2 stream between 650 and 750 °C (with a % ML less than 1.5 %) indicated that the carbon does not react with this chlorinating gaseous mixture.

These experimental evidences and assumptions (i.e. the phosphorus trichloride is volatilized; calcium chloride remained totally in the residue; carbon is consumed only for the carbochlorination reactions) permit using the percent mass loss for describing the reaction extent for temperatures lower than 775 °C. According to the data processing of Fig. 7, the Arrhenius' diagram is established for temperature range 625-750 °C and it is exhibited in Fig. 9. It depicts the evolution of the apparent carbochlorination rate (on

Conclusions

The dicalcium phosphate dihydrate (CaHPO 4 . 2H 2 O -brushite) is converted into calcium pyrophosphate (Ca 2 P 2 O 7 ) at temperatures higher than 400°C through monetite (CaHPO 4 ) as intermediate phase.

The thermodynamic predictions showed that the chlorination of the Ca 2 P 2 O 7 by only chlorine is not feasible for the temperatures up to 1100°C. The envisaged carbochlorination reactions of Ca 2 P 2 O 7 with (Cl 2 +CO and Cl 2 +C) are thermodynamically favorable in the whole temperature range investigated. The most stable chloride phases for phosphorus and calcium are PCl 3 , POCl 3 and CaCl 2 , respectively. The selectivity of the carbochlorination process is based on the wide difference on the vapor pressure of the chloride reaction products.

The reaction rate of the Ca 2 P 2 O 7 chlorination with Cl 2 +CO+N 2 at 950°C is weak due to the formation and melting of CaCl 2 creating a compact layer which prevents the access of reactant gases to the reaction zone.

Kinetics of the Ca 2 P 2 O 7 chlorination is enhanced at low temperatures and by using carbon instead of CO as a reducing agent. Carbochlorination reaction of calcium pyrophosphate is almost fully completed at 750 °C, leading to the separation of phosphorus chlorinated compounds from calcium chloride. A value of the apparent activation energy of about 104 kJ/mol characterizes the carbochlorination of Ca 2 P 2 O 7 in the temperature range from 625 to 750 °C.

The developed process can be envisaged as an alternative of the known wet and thermal methods for the beneficiation of the low grade phosphorus ores. Further, it can be used for the recycling and extraction of phosphorus from various wasted materials.

18

 18

Figure 1 .

 1 Figure 1. Behavior of the brushite samples (wet and dried sample) under nitrogen atmosphere heated up to 1000°C

Figure 2 .Figure 3 .

 23 Figure 2. Standard free energy change of the chlorination reactions of the Ca 2 P 2 O 7 with: (a): Cl 2 ; (b): Cl 2 +CO; (c): Cl 2 +C

Figure 4 .

 4 Figure 4. Evolution of the vapor pressure as a function of temperature for several chlorides generated during phosphate chlorination

Figure 5 .

 5 Figure 5. Evolution of the % mass loss of the sample as a function of time during carbochlorination of DCPI at 950 °C (a) and SEM-EDS analysis results (b)

21 Figure 6 .Figure 7 .

 2167 Figure 6. Results of the TG-DT analysis for the calcium chloride in nitrogen atmosphere

Figure 8 .

 8 Figure 8. SEM-EDS spectra of the (DCPI+C) initial sample and treatment residues obtained during carbochlorination by N 2 +Cl 2 at different temperatures

Figure 9 .

 9 Figure 9. Arrhenius diagram for the carbochlorination of the DCPI at 625-750 °C

  .

	Main phases of the raw samples are CaHPO 4 , CaHPO 4 . 2H 2 O and carbon. However, as mentioned in the Section 2, CaHPO 4 and CaHPO 4 . 2H 2 O are transformed into Ca 2 P 2 O 7 at temperatures higher than 400°C. The presence of the CaCl 2 . 2H 2 O in the carbochlorination residue generated at 625 °C is confirmed by XRD. One may mention that the identification of the hydrate species for CaCl 2 (CaCl 2 . 2H 2 O and CaCl 2 . 4H 2 O) is due to the hygroscopic character of the CaCl 2 which is transformed into its hydrate forms during sample preparation and XRD measurement.

Table 1 .

 1 Identified phases in the carbochlorination residues of DCPI at different temperatures logarithm scale) as a function of the inverse of the absolute temperature according to the kinetic equation and models described elsewhere[24,[START_REF] Erdem | [END_REF]. Note that the figures of the reaction rate deduced from % ML are corrected due to an apparent interaction of the silica sample boat with C+Cl 2 . A mean value of the apparent activation energy of about 104 kJ/mol is calculated from the data plotting of this Figure.Such a value tends to indicate that the overall reaction rate of the Ca 2 P 2 O 7 chlorination with Cl 2 +C is expected to be controlled by the chemical reaction.

	Temperature, Initial	°C	Main crystallized phases identified by XRD CaHPO 4 , CaHPO 4 . 2H 2 O, C*
	625		Ca 2 P 2 O 7 , CaCl 2 . 2H 2 O, C
	650		Ca 2 P 2 O 7 , CaCl 2 . 4H 2 O, CaCl 2 . 2H 2 O, C
	675		CaCl 2 . 4H 2 O, CaCl 2 . 2H 2 O, Ca 2 P 2 O 7 , C
	700		CaCl 2	. 2H 2 O, Ca 2 P 2 O 7 , C
	725		CaCl 2 . 2H 2 O, CaCl 2 , Ca 2 P 2 O 7 , C
	750		CaCl 2 . 2H 2 O, CaCl 2 , C
	775		CaCl 2 . 2H 2 O, CaCl 2 , C


James S P Macdonald

Patrick Cavanagh

Rufin Vanrullen
email: rufin.vanrullen@cerco.ups-tlse.fr

Attentional sampling of multiple wagon wheels

published or not. The documents may come   L'archive ouverte pluridisciplinaire

INTRODUCTION

The 'Wagon Wheel Illusion' is the appearance in movies of a wheel rotating in the opposite direction to its original motion. Such aliasing occurs when the wheel rotates at a slightly slower temporal frequency than the sampling rate of the camera recording it, since with each frame the wheel's spokes appear to have rotated backwards a small amount rather than forwards by a large amount. Surprisingly, a similar illusion can occur when viewing periodic, moving objects in the real world under continuous lighting conditions, such as sunlight [START_REF] Kline | Illusory motion reversal is caused by rivalry, not by perceptual snapshots of the visual field[END_REF][START_REF] Purves | The wagon wheel illusion in movies and reality[END_REF][START_REF] Schouten | Subjective stroboscopy and a model of visual movement detectors[END_REF][START_REF] Simpson | Illusory percepts of moving patterns due to discrete temporal sampling[END_REF][START_REF] Vanrullen | Attention-driven discrete sampling of motion perception[END_REF], leading many to suggest that motion perception may involve discrete sampling of visual information like a movie camera [START_REF] Andrews | The wagon-wheel illusion in continuous light[END_REF][START_REF] Crick | A framework for consciousness[END_REF][START_REF] Koch | The Quest for Consciousness: A Neurobiological Approach[END_REF][START_REF] Mccomas | The RULER model. Is this how the somatosensory cortex works[END_REF][START_REF] Purves | The wagon wheel illusion in movies and reality[END_REF][START_REF] Rojas | Do discreteness and rivalry coexist in illusory motion reversals?[END_REF][START_REF] Simpson | Illusory percepts of moving patterns due to discrete temporal sampling[END_REF][START_REF] Vanrullen | Is perception discrete or continuous?[END_REF][START_REF] Vanrullen | Attention-driven discrete sampling of motion perception[END_REF]. Specifically, it has been proposed that such discrete sampling originates in the attentional system [START_REF] Reddy | Neural correlates of the continuous Wagon Wheel Illusion: a functional MRI study[END_REF][START_REF] Vanrullen | The continuous Wagon wheel illusion and the 'when' pathway of the right parietal lobe: a repetitive transcranial magnetic stimulation study[END_REF][START_REF] Vanrullen | Attention-driven discrete sampling of motion perception[END_REF][START_REF] Vanrullen | The continuous wagon wheel illusion is associated with changes in electroencephalogram power at approximately 13 Hz[END_REF][START_REF] Vanrullen | The continuous Wagon Wheel Illusion is object-based[END_REF][START_REF] Vanrullen | The continuous Wagon Wheel Illusion depends on, but is not identical to neuronal adaptation[END_REF], since this 'continuous' Wagon-Wheel Illusion (c-WWI) is strongly associated with attention: it occurs far less frequently when attention is drawn away from the periodic stimulus by a concurrent demanding discrimination task [START_REF] Vanrullen | Attention-driven discrete sampling of motion perception[END_REF], it does not occur in the entire visual field [START_REF] Kline | Illusory motion reversal is caused by rivalry, not by perceptual snapshots of the visual field[END_REF], but rather in one object at a time that may be formed from several elements according to gestalt principles of association [START_REF] Vanrullen | The continuous Wagon Wheel Illusion is object-based[END_REF], and it has been associated with right parietal lobe processing with a variety of neurophysiological techniques (EEG: VanRullen et al., 2006;r-TMS: VanRullen et al., 2008;fMRI: Reddy et al., 2011). According to this proposal, the c-WWI is limited to attended stimuli because discrete sampling in the attentional system leads to aliasing in the attention-based motion perception system, but not in the lower-level motion perception system. Note that previous authors have proposed a distinction between either two [START_REF] Cavanagh | Attention-based motion perception[END_REF] or three [START_REF] Koch | The Quest for Consciousness: A Neurobiological Approach[END_REF] separate motion systems: both theories concurred on the distinction between a lower-level, pre-attentive stage and a higher-level, attention-based system, but differed on the possible existence of an intermediate stage.

An alternative explanation of the c-WWI is that it is a form of motion aftereffect that occurs while the stimulus is still present: a 'motion during-effect' caused by the spurious activation of low-level motion detectors after a period of adaptation [START_REF] Holcombe | Illusory motion reversal in tune with motion detectors[END_REF][START_REF] Holcombe | Illusory motion reversals from unambiguous motion with visual, proprioceptive, and tactile stimuli[END_REF][START_REF] Kline | Evidence against the temporal subsampling account of illusory motion reversal[END_REF][START_REF] Kline | Illusory motion reversal is caused by rivalry, not by perceptual snapshots of the visual field[END_REF][START_REF] Kline | Illusory motion reversal does not imply discrete processing: Reply to Rojas et al[END_REF]. With this adaptation view of the c-WWI, its association with attention is explained by proposing that attention regulates the amount of motion adaptation. Some other features of the c-WWI are also in line with such a proposal: it is a bistable percept that occurs sporadically, and it requires a period of viewing time before it is experienced [START_REF] Andrews | The wagon-wheel illusion in continuous light[END_REF][START_REF] Holcombe | Illusory motion reversal in tune with motion detectors[END_REF][START_REF] Kline | Illusory motion reversal is caused by rivalry, not by perceptual snapshots of the visual field[END_REF][START_REF] Kline | Illusory motion reversal does not imply discrete processing: Reply to Rojas et al[END_REF][START_REF] Vanrullen | The continuous Wagon Wheel Illusion depends on, but is not identical to neuronal adaptation[END_REF]. Other features still, however, are contradictory to this adaptation view, but are compatible with the proposal that it is a consequence of aliasing in at least one motion perception system due to discrete sampling: it has an optimal stimulus temporal frequency (of about 10 Hz) [START_REF] Vanrullen | Attention-driven discrete sampling of motion perception[END_REF], it does not depend on the stimulus' spatial frequency [START_REF] Purves | The wagon wheel illusion in movies and reality[END_REF][START_REF] Simpson | Illusory percepts of moving patterns due to discrete temporal sampling[END_REF][START_REF] Vanrullen | Attention-driven discrete sampling of motion perception[END_REF], it occurs for motion of both first-order stimuli (luminance-defined) and second-order stimuli (contrast-defined) [START_REF] Vanrullen | Attention-driven discrete sampling of motion perception[END_REF], even though these two types of stimuli may be processed by different areas of the brain [START_REF] Dumoulin | Cortical specialization for processing first-and second-order motion[END_REF]Vaina & Soloviev, 2004, although see Nishida, Sasaki, Murakami, Watanabe, & Tootell, 2003;[START_REF] Seiffert | Functional MRI Studies of Human Visual Motion Perception: Texture, Luminance, Attention and After-effects[END_REF], and its strength decreases with eccentricity, whereas both the static and flicker motion aftereffects increase with eccentricity [START_REF] Vanrullen | The continuous Wagon Wheel Illusion depends on, but is not identical to neuronal adaptation[END_REF], indicating that it is not solely driven by adaptation. Furthermore, the bistability of the c-WWI can be reconciled with the notion of discrete attentional sampling by proposing that adaptation leads to a competition for dominance between the veridical percept generated by the lower-level motion perception system, which is unaffected by discrete sampling, and its alias from the attention-based system where the c-WWI originates. The role of adaptation in this proposal is therefore in varying the relative strengths of the veridical percept and its alias over time.

One way in which the discrete attentional sampling and adaptation explanations of the c-WWI might be distinguished is by investigating the effect of dividing attention between multiple periodic motion stimuli. The discrete attentional sampling hypothesis predicts that the optimal stimulus temporal frequency for observing illusory reversals of the direction of motion would decrease as the number of attended stimuli increased. This is because each individual stimulus would be sampled in turn, reducing the rate at which any one stimulus is sampled (unless sampling is assumed to be purely parallel and resource-unlimited, a possibility that we address below). The adaptation hypothesis, on the other hand, has no reason to predict a change in optimal stimulus temporal frequency. Instead, it could predict a decrease in illusion strength when attention is divided among multiple stimuli, since adaptation is known to depend on attention [START_REF] Chaudhuri | Modulation of the motion aftereffect by selective attention[END_REF][START_REF] Rees | Modulating Irrelevant Motion Perception by Varying Attentional Load in an Unrelated Task[END_REF], and the overall amount of attention allocated to each individual target likely decreases with set size.

The expected outcome of the set size manipulation we propose depends on the purported mechanism of divided attention. It is still debated whether attention divides in a sustained fashion to subsume multiple objects (a parallel model), or whether it shifts sequentially between the different simultaneously presented targets (a serial model). A review of the vast and ongoing literature concerning the mechanism of divided attention is beyond the scope of this article, but for a recent review see [START_REF] Jans | Visual spatial attention to multiple locations at once: the jury is still out[END_REF]. For our purposes, a serial model would imply that each target is sampled in only a subset of the attentional 'snapshots', as the focus of attention shifts continually between targets. A snapshot of each target would be stored in memory and compared with subsequent snapshots, in order to extract the motion signal for each target individually. This reduction in the sampling rate of individual targets would result in a corresponding reduction in the optimal temporal frequency at which the c-WWI occurs.

Likewise, a parallel model in which all targets are sampled simultaneously, but with a sampling rate dependent on the complexity of each sample, could make a similar prediction: a decrease of optimal c-WWI temporal frequency with increasing set size. Only a strict parallel model, in which all targets are sampled simultaneously and at the same rate regardless of set size, would predict a constant c-WWI optimal temporal frequency. Since this is the same prediction as the adaptation interpretation, such an outcome would not allow us to differentiate the competing theories.

The relation of c-WWI optimal frequency to attentional sampling rate In order to interpret the results of our set-size manipulation, it is important to understand the mathematical relationship between the frequency tuning of the c-WWI and the rate of discrete attentional sampling that we propose causes it. An earlier paper [START_REF] Vanrullen | Attention-driven discrete sampling of motion perception[END_REF] describes why a maximal c-WWI at 10 Hz implies that the underlying sampling rate is approximately 13 Hz. The explanation is as follows: The strongest motion signal from a periodic stimulus occurs when the motion steps are 1/4 cycle. Therefore the maximum reverse motion strength will occur when the stimulus steps 3/4 of a cycle with each sample. For sampling of a 10 Hz signal to fall at the 3/4 cycle point on each sample, the sampling rate must be 4/3 of 10 Hz -13.3 Hz. Currently, we only have an estimate of the sampling rate of attention when a single periodic motion stimulus is attended at a time. [START_REF] Kline | Illusory motion reversal is caused by rivalry, not by perceptual snapshots of the visual field[END_REF] did use two simultaneous wagon wheel stimuli, but did not manipulate temporal frequency or measure the optimal temporal frequency of the illusion. We propose to quantify the temporal-frequency tuning of the c-WWI as a function of the number of attended wagon-wheel stimuli. If attention moves from wheel to wheel to capture samples, then the sampling rate should decrease as the number of stimuli increases.

Our study

We asked participants to report the number of motion direction 'reversal events' (i.e., brief reversals of the direction of motion) occurring in a set of slightly ambiguous, rotating stimuli and we varied both the set size (i.e., the number of stimuli presented simultaneously) and the rotation temporal frequency, from trial to trial. Each individual stimulus was composed of two superimposed gratings rotating in opposite directions, with one at a higher contrast than the other so that the perceived motion was biased in that direction. This slightly ambiguous, periodic stimulus was found to be more often seen as reversing. To encourage participants to report all perceptual reversals, several actual motion direction reversals occurred during each 40-second trial; whenever the number of reported reversals was greater than the actual number, we inferred that the difference was the number of illusory reversal events (i.e., instances of the c-WWI) the participant had seen in that trial. We plotted this measure against temporal frequency for each set size, and fit the data points with a Gaussian curve, in order to assess the temporal-frequency tuning of the c-WWI across set sizes. METHOD Participants 12 participants were recruited at the Université Paul Sabatier in Toulouse and were paid for their participation. Three participants were excluded and replaced because they reported very few illusory reversal events. Another three were excluded and replaced because their mean correlation of reported reversals and real reversals across temporal frequencies was below 0.5 (i.e., they were not accurately reporting the number of real reversal events, and hence not performing the task properly; see Procedure for more details of this exclusion criterion). The age range of those included was 21 to 34 years (M = 27.2 years, SD = 3.7 years), and there were eight males. For the motion after-effect control experiment, four participants were recruited at the University of Oxford, having an average age of 28.3 years (SD = 3.8 years), and including one male. All of the participants had normal or corrected-to-normal vision.

Apparatus and Stimuli

The experiments were created and run with Matlab 2007a (The Mathworks, Inc., 2007)

on a

Windows PC attached to a 17" monitor at a resolution of 1024 x 768 and a refresh rate of 100 Hz. A viewing distance of 57 cm was maintained throughout the experiment. In each trial, one to four clockwise rotating circular gratings of 1.4° radius were presented equally spaced but at a randomly determined location on an imaginary circle of 2.5° radius that was centred at fixation (nearest contours 2.3°, 1.6°, and 0.9° apart for set sizes two to four, respectively). The circular gratings were contained within a disc of 4.7° radius that was mid-grey (30.3 cd/m 2 ), and the background of the display was black. A red circle of 0.1° radius at the centre of the screen served as a fixation point (see Figure 1 for example displays). Each circular grating was in fact composed of two superimposed gratings moving in opposite directions. The contrast difference between the two was adjusted for each participant pre-experimentally so as to make rotation direction weakly biased. Each circular grating had a sinusoidal luminance modulation of eight cycles per rotation. The grating(s) rotated during each 40-s trial at a constant temporal frequency that was selected at random from one of six possibilities: 2.0, 4.0, 5.9, 7.7, 10.0, and 12.5 Hz. In each trial, two to five real reversals of the direction of motion occurred at randomly selected intervals in a subset or in all of the gratings, lasting between 0.5 and 1.5 s before reverting to the original direction of motion. The minimum time between these real reversal events was 1.5 s and none occurred during the first 3 s of a trial. A counterbalanced set of 192 different stimulus displays consisting of six temporal frequencies, four set sizes, and four amounts of real reversal events, was presented in four blocks of 48 trials. Each block contained all temporal frequencies and all set sizes, but only two amounts of real reversal events that were counterbalanced across blocks, such that each block contained either two and four, two and five, three and four, or three and five real reversal events. The order in which these pairs of numbers of real reversal events were presented was counterbalanced across participants.

In the motion after-effect control experiment, the same rotating circular gratings were presented for 20 s per trial, at a set size of either one or four, with no direction reversals. The same six temporal frequencies were used. At the end of each trial, the circular gratings stopped rotating, and a large red circle appeared around them to emphasise this event. In half of the blocks, the circular gratings remained motionless after stopping rotating -these blocks measured the 'static' motion after-effect; in the other half of the blocks the circular gratings counterphased (flickered) at the same temporal frequency that they had previously been rotating -these blocks measured the 'flicker' motion after-effect (e.g., [START_REF] Nishida | Motion aftereffect with flickering test patterns reveals higher stages of motion processing[END_REF], see Mather, Pavan, Campana, & Casco, 2008 for a review). A counterbalanced set of 24 different stimulus displays consisting of six temporal frequencies, two set sizes, and two motion directions was presented in each of four blocks. Blocks alternated between measuring the static and flicker motion after-effects.

Procedure

The contrast difference between each pair of superimposed gratings was adjusted for each participant pre-experimentally so as to be somewhat ambiguous in rotation direction but not indistinguishable. This was achieved by displaying a series of these biased motion stimuli (pairs of superimposed gratings) of increasing contrast difference for 10 seconds each, and asking the participant if the direction of motion was clear and if so, whether it was clockwise or anticlockwise. The first contrast difference at which the participant was able to state the correct direction of motion with confidence was selected to be used for the first block of 48 trials. Next, each participant completed a short practice block consisting of eight randomly-selected trials before starting the experiment proper. These practice trials were identical to the trials in the experimental blocks, which proceeded as follows (see Figure 1): each trial's stimulus display lasted 40 s, after which the rotating gratings stopped moving and above them in white the words, 'How many reversal events did you see? Enter digit', appeared. The participants were instructed to report the number of reversal events they had seen during the trial at this point by pressing one of the keys on the numeric keypad (0-9). If they had seen more than nine reversal events they were instructed to report nine. A reversal event was defined as a brief reversal of the direction of motion in one or more stimuli followed by a return to the original direction of motion. Responses were not speeded and no feedback was given. Upon response the next trial began. After each 48-trial block the contrast difference between the two superimposed gratings (and thus, the ambiguity of the motion direction) could be raised or lowered in order to adjust task difficulty, depending on whether the participant had seen very many or very few illusory reversal events across all temporal frequencies.

In order to verify that participants were performing the reversal event counting task and were attending effectively to all the gratings, we calculated the Pearson correlation of the number of reported reversal events with the number of real reversal events for each participant, separately for each of four blocks and six temporal frequencies. If the average correlation across temporal frequencies for any block was below 0.5 the block was excluded from the analysis.

With this criterion, three participants were excluded from the analysis, since all of their blocks failed, and were replaced with three new participants; five participants had two blocks excluded, and one participant had one block excluded. The remaining six participants had no blocks excluded. The mean correlation across participants after exclusion of blocks was 0.66 (SD = 0.10). Note that the occurrence of the illusion is bound to affect this correlation value; however, given a particular illusion strength, on average more actual reversals should always result in more reported reversals if participants are performing the counting task correctly. Thus, the presence of a positive correlation is a valid marker of reversal-counting performance.

In the motion after-effect control experiment, participants were instructed to observe the rotating gratings whilst maintaining fixation, and when they had stopped rotating, wait until the motion after-effect (illusory motion in the opposite direction) had stopped before pressing the '0' key on the numeric keypad. Participants pressed the 'S' key to initiate each trial and there was a four-second inter-trial interval.

Figure 1. Experimental Procedure. Each trial display contained one to four circular gratings rotating clockwise with some ambiguity in the rotation direction introduced by superimposing two counter-phased gratings of opposite directions and slightly different contrasts. Each trial lasted for 40 s, during which time two to five actual 'reversal events' (lasting between 0.5 and 1.5 s) occurred in a subset or all of the gratings. A 'reversal event' was defined as a brief, single reversal of the direction of motion simultaneously involving any number of gratings, followed by a return to the original direction of motion. There was a delay of at least 1.5 s between reversal events. Participants were instructed to report the number of reversal events they had observed at the end of each trial.

RESULTS

Continuous Wagon-Wheel Illusion temporal-frequency tuning

Our experimental procedure gave us, for each of four blocks per participant, the number of reported reversal events in each trial to compare to the number of actual reversal events. We considered overestimations of the number of reversals as manifestations of the c-WWI, and for each participant we calculated the average ratio of reported reversal events to actual reversal events across trials as a function of set size and temporal frequency. After averaging these ratios across participants, we fitted the temporal-frequency tuning of the illusion for each set size with a Gaussian function (Figure 2) with four parameters: mean (μ), standard deviation (σ), amplitude (A), and baseline (b). The fits minimized the error term |ξ| in the following equation:

2 2 ( ) 2 ( ) ( ) 2 f A rr f e b f µ σ ξ πσ -- = + +
where f is the temporal frequency of the stimulus and rr is the 'reversal ratio' between reported and actual reversal events. Recall that the attentional sampling interpretation of the c-WWI predicts a drop in temporal-frequency tuning with increasing set size, whereas the adaptation interpretation a priori does not. The data points for each set size were fitted with a Gaussian function in order to determine the optimal temporal frequency of the c-WWI (represented by a red vertical arrow in each case).

The optimal temporal frequency decreased significantly with increasing set size (p < .001).

The data points for each set size fit a Gaussian function well, the mean of which represents the stimulus temporal frequency most likely to induce an illusory reversal of the direction of motion (the optimal temporal frequency of the c-WWI). For set size one, the mean of our fitted Gaussian, and hence the optimal temporal frequency, was 9.2 Hz, a figure very close to the optimal temporal frequency found previously for a single periodic rotating stimulus, 10 Hz [START_REF] Vanrullen | Attention-driven discrete sampling of motion perception[END_REF]. Our data for set size one therefore replicate this previous finding and, importantly, also confirm that our new c-WWI paradigm engages similar perceptual mechanisms to the paradigms in previous work on the c-WWI [START_REF] Vanrullen | Attention-driven discrete sampling of motion perception[END_REF].

With regard to the effect of set size on optimal temporal frequency, inspection of Figure 2 suggests that optimal temporal frequency decreased consistently with increasing set size, from approximately 10 Hz for set size one to just over 5 Hz for set size four. In order to test these data statistically, we fitted the temporal-frequency tuning of the illusion for each set size with a Gaussian function in the same way as before but for each participant individually. The means of these Gaussian fits (i.e., the illusion optimal temporal frequency) were entered into a repeated measures one-way analysis of variance (ANOVA) with set size as a factor that revealed a significant decrease of optimal temporal frequency with increasing set size, F(3, 33) = 8.65, MSE = 16.27, p < .001, ηp 2 = .44 (two tailed, as are all further statistical tests). These data are shown in Table 1 and are plotted in Figure 3(a). Pairwise comparisons revealed that optimal temporal frequency for set size one was significantly higher than that for set sizes two, three, and four (all p < .005). Our data therefore suggest that attending to multiple objects reduces the rate of sampling of each individual object compared to the rate of sampling for a single attended object. This finding is in line with the prediction derived from the discrete attentional sampling interpretation of the c-WWI.

We performed the same ANOVA analysis that we had used on the mean for the other parameters of the Gaussian fits (i.e., standard deviation, amplitude, and baseline -all shown in Table 1). None of these three ANOVA were significant, however, therefore our data do not indicate an effect of set size on temporal-frequency tuning standard deviation, F(3, 33) = 1.55, MSE = 4.44,p = .219,amplitude,F(3,33) = 1.79,MSE = 65.15,p = .169,or baseline,F(3,33) = 1.29, MSE = 0.094, p = .295, none of which we expected to find.

Continuous Wagon-Wheel Illusion strength

One prediction that could be derived from the adaptation interpretation of the c-WWI is that illusion strength would decrease with increasing set size, since the amount of adaptation depends on the allocation of attention, which in turn depends on the number of simultaneously attended objects. We took the peak reversal ratio derived from the Gaussian fit for each set size as a measure of the strength of the illusion (presented in Figure 3b), and performed a one-way ANOVA on peak reversal ratio with set size as the factor. This analysis did not provide evidence that c-WWI strength varied with set size, F < 1. This result contrasts with the decrease in optimal temporal frequency with increasing set size (presented in Figure 3a). Motion after-effect temporal-frequency tuning

Although the adaptation interpretation of the c-WWI would a priori predict no change in temporal-frequency tuning with set size, one a posteriori explanation could be that adaptation of different populations of neurons, which are differentially tuned to rotation temporal frequency, is responsible for the effect of set size on optimal temporal frequency. For example, observation of a single rotating circular grating might activate motion-sensitive neurons with large receptive fields tuned to higher temporal frequencies, whereas observation of multiple rotating circular gratings might instead involve neurons with smaller receptive fields tuned to lower temporal frequencies. To evaluate this interpretation of our results, we carried out a second experiment in which participants viewed either one or four circular gratings rotating at the same six temporal frequencies as in the main experiment and reported the duration of the motion after-effect (MAE) after the gratings had stopped rotating and either remained motionless (static MAE) or counterphased (flicker MAE). As can be observed in Figure 4, the static and flicker MAE duration showed no significant variation with temporal frequency and this absence of tuning was unaffected by set size. This was confirmed by two-way ANOVA on MAE duration with set size and temporal frequency as factors: F < 1 for the interaction of set size and temporal frequency for both the static and flicker MAE. There was also no main effect of temporal frequency, both F < 1. There was no main effect of set size on the duration of the flicker MAE, F < 1, although there was for the static MAE, F(1, 3) = 11.09, MSE = 2.02, p = .048, ηp 2 = .79, with longer MAE durations for the larger set size. This effect could be a result of the larger overall adapted area. Nevertheless, since the temporal-frequency tuning of the MAE was unaffected by set size, this experiment does not support an adaptation interpretation of the relation between the c-WWI optimal temporal frequency and set size. Motion after-effect duration temporal-frequency tuning did not vary with set size, ruling out an explanation of the c-WWI optimal temporal frequency set size effect in terms of adaptation of different populations of motion-sensitive neurons in the different set size conditions.

DISCUSSION

The stimulus temporal frequency most likely to induce an illusory reversal of the direction of motion for set size one was very similar to that found previously for a single periodic rotating stimulus [START_REF] Vanrullen | Attention-driven discrete sampling of motion perception[END_REF]. This figure, 10 Hz, corresponds to an attentional sampling rate of 13.3 Hz (see Introduction and ibid.). The data for set size one therefore reinforce the claim that attention samples visual information discretely at around 13 Hz [START_REF] Vanrullen | Attention-driven discrete sampling of motion perception[END_REF][START_REF] Vanrullen | The continuous wagon wheel illusion is associated with changes in electroencephalogram power at approximately 13 Hz[END_REF].

Our manipulation of set size was intended to distinguish between the discrete attentional sampling and adaptation interpretations of the c-WWI, since each makes a distinct prediction about the effect of set size. The discrete attentional sampling interpretation leads to the prediction that optimal temporal frequency would reduce as the number of attended stimuli increases, since the sampling rate of each individual stimulus would be a proportion of the sampling rate of a singularly attended stimulus. The adaptation interpretation, on the other hand, gives rise to no a priori rationale for a change in optimal temporal frequency with set size.

Instead, a decrease in illusion strength with increasing set size might be predicted, since each of multiple stimuli would receive less attention than a lone attended stimulus, and adaptation is known to scale with attention [START_REF] Chaudhuri | Modulation of the motion aftereffect by selective attention[END_REF][START_REF] Rees | Modulating Irrelevant Motion Perception by Varying Attentional Load in an Unrelated Task[END_REF]. We found that for set sizes two, three and four, in which attention was divided between stimuli, optimal temporal frequency decreased compared to set size one and illusion strength remained unchanged, in line with the discrete attentional sampling interpretation of the c-WWI, and in opposition to the predictions of the adaptation interpretation.

It is difficult to reconcile the finding that optimal temporal frequency decreases with set size with the adaptation interpretation of the c-WWI. According to this account, illusory reversals occur due to the spurious activation of low-level motion detectors after a period of adaptation [START_REF] Holcombe | Illusory motion reversal in tune with motion detectors[END_REF][START_REF] Holcombe | Illusory motion reversals from unambiguous motion with visual, proprioceptive, and tactile stimuli[END_REF][START_REF] Kline | Evidence against the temporal subsampling account of illusory motion reversal[END_REF][START_REF] Kline | Illusory motion reversal is caused by rivalry, not by perceptual snapshots of the visual field[END_REF][START_REF] Kline | Illusory motion reversal does not imply discrete processing: Reply to Rojas et al[END_REF], and the optimal temporal frequency of 10 Hz of the c-WWI is the consequence of the tuning of low-level motion detectors (Reichardt detectors, Reichardt, 1961) being concentrated around 10 Hz [START_REF] Kline | Illusory motion reversal does not imply discrete processing: Reply to Rojas et al[END_REF], since that is the maximal frequency range of normal human motion sensitivity [START_REF] Snowden | Temporal frequency filters in the human peripheral visual field[END_REF]). An account of our results in terms of such motion detectors would therefore have to surmise that an increase in the load on attention, for example, the number of attended objects, results in the spurious activation of a different subset of these detectors that are tuned to temporal frequencies lower than 10 Hz. For example, observation of a single rotating circular grating might activate motion-sensitive neurons with large receptive fields tuned to higher temporal frequencies, whereas observation of multiple rotating circular gratings might instead engage neurons with smaller receptive fields tuned to lower temporal frequencies. We did not find support for such a hypothesis in our motion after-effect control experiment, in which the temporal-frequency tuning of MAE duration did not vary with set size. We suggest it is more parsimonious to interpret our results in terms of discrete attentional sampling. Furthermore, since set size effects are considered a hallmark of attentional involvement, our results bolster support for the discrete attentional sampling explanation of the c-WWI in general.

We also considered whether our results could be explained by the occurrence of saccades during the observation of multiple rotating stimuli: if participants had ignored the instruction to maintain their gaze on the fixation point at the centre of the screen, they may instead have shifted their gaze between stimuli at set sizes two and above. Had it occurred, the effect of this intermittent input may have generated additional aliasing artefacts which could have contributed to the downward shift in c-WWI optimal temporal frequency from around 10 Hz at set size one to 5-7 Hz at larger set sizes. There are problems with this interpretation, however, that render it unlikely in our opinion. Eye movements can occur at up to 5-6 Hz, or every 160-200 ms [START_REF] Rayner | Eye movements in reading and information processing: 20 years of research[END_REF]. Therefore the simplest hypothesis is that regular shifts of fixation would have generated a secondary peak in illusory reversals at temporal frequencies below 3-4 Hz -about three quarters of eye movement frequency (see the introduction for an explanation of why this would be so). We did not observe any such bimodality in our data. A refined hypothesis could be that several additional peaks were generated, not just at the eye movement switching frequency, but also at harmonic multiples of this frequency. These multiple peaks within the range we presented (2-12.5 Hz) were also not visible in our data. Finally, the possibility that these multiple peaks could have merged and induced a general baseline increase for set sizes two and above compared to set size one is contradicted by the absence of a significant baseline effect (see Table 1).

We note that in one recent study [START_REF] Holcombe | Splitting attention reduces temporal resolution from 7 Hz for tracking one object to <3 Hz when tracking three[END_REF]) the authors reached a similar conclusion to our own -that the sampling rate of attention decreases when the number of attended targets increases -in the context of multiple-object tracking. Holcombe and Chen reported that the ability to track objects situated on rotating concentric circles was determined by the temporal frequency of the circles. Critically, as the number of objects to be tracked increased, the maximum temporal frequency at which tracking could be performed reliably decreased. The authors highlighted that only a serial model of attention to multiple objects would a priori predict such an effect, whereas a parallel model would not.

Despite the evidence favoring the discrete attentional sampling proposal, there is one aspect of the results that requires an amendment to the simplest version of sampling: the effect of set size was smaller than would be predicted by a purely serial mechanism of divided attention. Specifically, the optimal temporal frequency of the illusion did not drop linearly with the reciprocal of the number of items sampled (if it had, we would have obtained optimal temporal frequencies of around 5, 3.3, and 2.5 Hz for set sizes two, three, and four, respectively). On the other hand, our data are also incompatible with a strict parallel and resource-unlimited model, since this model would predict no change in optimal temporal frequency with progressive increases in set size. Therefore, a more complex model of attentional deployment must be invoked to explain our results. We propose three such models: firstly, a modified parallel strategy, in which the sampling rate of attention decreases when the contents of attention are more complex (e.g., when more than one object must be attended); secondly, an irregular serial strategy, in which attentional sampling rate is maintained at approximately 13

Hz, but a variable number of samples are taken from each successively attended object; thirdly, a faster serial strategy, in which individual objects are selected sequentially, but sampling rate increases with set size.

Although we are not able to further disentangle these three models of attentional deployment with the current data, it may be possible to do so in the future by repeating the experiment while recording participants' EEG along the same lines as the study conducted by [START_REF] Vanrullen | The continuous wagon wheel illusion is associated with changes in electroencephalogram power at approximately 13 Hz[END_REF], in which EEG power in the right parietal lobe at 13 Hz changed with the onset and offset of illusory reversals of the direction of motion: the modified parallel strategy would predict that EEG signatures of the c-WWI would occur at lower frequencies as set size increased, the irregular serial strategy would predict no influence of set size on the frequency of the EEG correlate, and the faster serial strategy would predict changes in the EEG power spectrum at successively higher frequencies as set size increased.

The three models we have proposed are necessarily based on simplifying assumptions.

For example, we assume a uniform attentional mechanism across space. On the other hand, it is possible that distinct attentional mechanisms exist in the two hemispheres [START_REF] Alvarez | Independent resources for attentional tracking in the left and right visual hemifields[END_REF]. Our experimental procedure, however, does not allow us to investigate the potential effects of this factor, since perceived motion reversals cannot be unambiguously attributed to one hemisphere or the other.

To summarise, we appraised the predictions of two competing interpretations of the c-WWI, namely discrete attentional sampling and adaptation, in a situation where multiple periodic stimuli are attended simultaneously. The optimal temporal frequency of the illusion decreased with increasing set size whereas illusion strength did not, in line with the discrete attentional sampling interpretation of the c-WWI and contrary to the adaptation account.

Figure 2 .

 2 Figure 2. Mean reversal ratio (reported/actual) as a function of circular grating rotation temporal

Figure 3 .

 3 Figure 3. (a) c-WWI optimal temporal frequency (i.e., the grand average of the means of the

Figure 4 .

 4 Figure 4. Grand average motion after-effect duration for static (left column) and flicker (right

Table 1

 1

	Set Size Mean (μ, in Hz)	Stand. Dev. (σ, in Hz)	Amplitude (A)	Baseline (b)	Peak Rev. Ratio
		M	SD	M	SD	M	SD	M	SD	M	SD
	1	8.61	0.76	3.76	0.71	9.91	3.16	0.84	0.12	1.76	0.16
	2	6.58	0.80	3.82	0.65	7.37	2.42	0.95	0.11	1.73	0.12
	3	6.38	0.65	2.88	0.30	7.55	2.51	0.93	0.11	1.84	0.10
	4	6.01	0.60	2.63	0.23	4.24	1.05	1.06	0.04	1.73	0.14

. Grand mean (and SEM) of Gaussian fit parameters (mean, standard deviation, amplitude, and baseline) performed individually for each participant, and peak reversal ratio, as a function of set size.

ACKNOWLEDGEMENTS

This research was funded by a EURYI Award to Rufin VanRullen.