Caroline Chubilleau

Bertrand Lenoir

Christophe Candolfi

Philippe Masschelein

Anne Dauscher
email: anne.dauscher@univ-lorraine.fr

Emmanuel Guilmeau

Claude Godart

Thermoelectric properties of In0.2Co4Sb12 skutterudites with embedded PbTe or ZnO nanoparticles

Keywords: thermoelectric materials, C: electrical transport, C: thermal transport, C: microstructure

come

Introduction

The development of novel highly-efficient thermoelectric materials for solid-state energy conversion -refrigeration and electrical generation from waste heat -is currently at the core of worldwide investigations. The efficiency of a thermoelectric material is governed by the dimensionless figure of merit ZT defined as where S is the Seebeck coefficient (or thermopower), ρ is the electrical resistivity and T is the absolute temperature [START_REF] Rowe | Thermoelectrics Handbook: Macro to Nano[END_REF]. In this equation, the total thermal conductivity κ is the sum of the electrical e  and the lattice contributions l  . ZT improvement is thus obtained by simultaneously achieving low thermal conductivity, low electrical resistivity and high thermopower. The main obstacle to overcome is related to the fact that the thermopower is inversely related to the electrical conductivity according to the Boltzmann transport equation and, as a result, maximisation of one cannot be achieved without minimisation of the other [START_REF] Ure | Thermoelectricity : Science and Engineering[END_REF].

Skutterudites such as CoSb 3 are considered as thermoelectric materials of great interest due to their good electrical properties. Nevertheless, their too high thermal conductivity constitutes a limiting factor to achieve ZT values beyond unity [START_REF] Uher | Skutterudites: prospective novel thermoelectrics[END_REF][START_REF] Sales | [END_REF]. Among the strategies used to optimize their thermoelectric properties, atomic substitutions on the Co or Sb sites and/or single or even multiplefilling were extensively studied [5, and refs therein]. More recently, bulk nanostructuring, associated or not with the two previous means, has also steered their performance towards significant improvement.

Indeed, this interesting way allows increasing phonon scattering on the numerous interfaces thereby lowering the thermal conductivity. Different kinds of nanostructured compounds have been on focus:

consolidated nanopowders [START_REF] Toprak | [END_REF][7][8][9][10], quenched-in nanostructures [START_REF] Uher | Recent advances in the development of efficient n-type skutteudites[END_REF][START_REF] Tang | High performance nanostructured thermoelectric materials prepared by melt-spinning and spark plasma sintering[END_REF][START_REF] Li | [END_REF][13][14], mixed nano-micron sized powders [15][16][17][18][19][20][21][22][23][24][25][26][27][28] or in-situ formed nanostructures [29][30][31][32][33].

In prior studies, we reported the effect on thermoelectric properties of semiconducting PbTe 25,26 or insulating ZnO 27,28 nano-sized particles embedded in a micron-sized CoSb 3 matrix.

While both types of inclusions were found beneficial near room temperature, their influence at higher

) ( 2 l e T S ZT     
temperatures was more elusive. This last result was due to a change in conduction type for PbTe (from p to n in the absence and presence of PbTe, respectively) and to large agglomeration of the nanoparticles at the grain boundaries for ZnO.

Here, we extend this approach to an optimized single-filled skutterudite In 0.2 Co 4 Sb 12 (ZT value of 1.05 at 700 K [34]) and investigate the influence on the transport properties of the presence of PbTe and ZnO nanoparticles. The paper is organized as follows. After presenting the experimental details, we discuss the results of a theoretical model developed with the aim of studying the influence of nanoparticles on the thermal transport. The analysis of the structural and chemical characterizations is followed by the results of the electrical and thermal properties and eventually, the dimensionless thermoelectric figure of merit.

Experimental procedures

Synthesis of the materials

Polycrystalline samples of nominal composition In 0.4 Co 4 Sb 12 were prepared by a combination of melting and solid-state reactions [START_REF] Da Ros | Proc. Mater. Res. Soc. Symp[END_REF][START_REF] Leszczynski | Proc. 6 th European Conf. Thermoelectrics[END_REF]. Stoechiometric amounts of high-purity In shots (99.99%), Sb shots (99.999%), and Ni-free Co (99.998%) powders were loaded into quartz ampoules in an argonatmosphere glovebox. The tubes were sealed under He/H 2 atmosphere, placed in a vertical rocking furnace and heated up to 650°C at a rate of 1°C/min. After a dwell time of 84 h at this temperature, the ampoules were finally quenched in a room-temperature water bath. The obtained ingots were then ground in an agate mortar into fine powders (<100 μm) that were cold-pressed into pellets. To ensure a good chemical homogeneity of the skutterudite phase, the pellets were annealed in a quartz ampoule sealed under He/H 2 atmosphere at 650°C for 120 h. Prior to the densification process, the annealed pellets were powdered again (<50 µm). The same batch was used to prepare both series of composites (ZnO and PbTe).

The ZnO nanoparticles were purchased from Alfa Aesar with a size of about 100 nm as measured by transmission electron microscopy. The PbTe nanoparticles were prepared by laser fragmentation in water from PbTe micron-sized particles according to the method described in Ref. [START_REF] Chubilleau | [END_REF]. The mean size of the spherically-shaped nanoparticles is about 6 nm in diameter. The x wt% PbTe and ZnO (0 ≤ x ≤ 12) composites were prepared via a liquid route from a stoichiometric mixture of micron-sized skutterudite powders and PbTe or ZnO nanoparticles. Water was used as the dispersing medium in the presence of PbTe while a methanol/hexane (3:1) mixture at 0.025 M was used with ZnO [27,28]. The as-dispersed compounds in the liquid (20 min in an ultrasonic bath) were then immediately immerged in liquid nitrogen and dried by freeze-drying during 12 hours (FTS-EZ585 system, 17 mTorr, -80°C). Finally, the composite powders were densified by spark plasma sintering (SPS Dr Sinter 515S) in graphite dies (10.4 mm in diameter) at 600°C during 6 min under 50 MPa. The relative densities of the specimens (Table 1) were finally evaluated from their dry mass and geometrical dimensions with respect to their theoretical densities, d th , calculated using Eq. ( 2 (2)

Materials characterization and transport properties measurements

Structural characterizations were performed by powder X-ray diffraction (PXRD) on a Bruker D8 Advance diffractometer using Co Kα 1 radiation in a 2θ-range of 25°-60°. The PXRD patterns were analyzed by the Rietveld method implemented in the Fullprof software [38] using the structural parameters of CoSb 3 as a starting model. The microstructure of the samples was characterized by scanning electron microscopy (SEM) on a Philips FEG XL 30. Electron probe microanalysis (EPMA) was carried out to check the chemical homogeneity and composition of all the samples obtained after the SPS treatment. The actual compositions of the skutterudite matrix were In 0.20 Co 4 Sb 11.79 and In 0.20 Co 4 Sb 11.86 for the ZnO-and PbTe-containing samples and did not show significant variations with the amount of nanoparticles introduced (these chemical formulae were normalized to four cobalt atoms).

Hereafter, the samples will be labelled using this actual composition.

Low-temperature thermoelectric properties (2-300 K) were measured simultaneously using a physical properties measurement system (PPMS, Quantum Design) with the thermal transport option TTO under zero magnetic field. The bar-shaped samples (of typical dimensions 2.5×2.5×9 mm 3 ) were cut perpendicularly to the press direction. The copper bars used as thermal and electrical contacts were attached onto the samples using BiSn braze. Hall coefficient measurements were conducted on the same samples between 5-300 K by sweeping the magnetic field between -3 and +3T with the ACT option of the PPMS. A five-probe method was used to carry out these measurements.

High-temperature electrical resistivity and thermopower were simultaneously measured between 300 and 700 K using an ULVAC-ZEM3 apparatus. The total thermal conductivity was calculated from the thermal diffusivity D measured between 300 and 700 K with the laser flash technique (NETZSCH LFA 427), the specific heat C p (NETZSCH DSC 404 F3 Pegasus), and the density d using Eq.(3).

d C D p     (3)

Results and discussion

Theoretical model

In order to get insights into the influence of both the nanoparticles size and the volume fraction on the thermal transport, we tried to model the lattice thermal conductivity of the In-filled skutterudite taking into account the presence of nanoparticles. Within the Debye model, is written as

k l k r = k B 2p 2 n k B ae ø ÷ 3 T 3 t (x,T )
x 4 e x e x -1 ( )

2 0 q D T dx (4)
where is the Boltzmann constant, is the sound velocity, is the reduced Planck constant, (x,T) is the relaxation time of phonons and is a dimensionless parameter with  the phonon pulsation. If we adopt the Matthiessen rule, (x,T) is the sum of several mechanisms assumed to be independent from each other:

t -1 = v L + Aw 4 + Bw 2 Te -q D 3T + Cw 2 (w 0 2 -w 2 ) 2 (5)
In Eq.( 5), the first term represents phonon scattering by grain boundaries with the mean grain size, the second term is related to point defect scattering (Rayleigh diffusion), the third term is associated to phonon-phonon scattering (Umklapp processes) and the last term stands for the interaction between the vibrational normal modes of the structure and the modes of the In atoms considered as localized (Einstein oscillators). In this last term,  0 is the characteristic vibrational pulsation of the In atoms. The presence of nanoparticles is expected to disrupt the heat-carrying phonons giving rise to an additional relaxation time. We followed here the model developed by Mingo et al. [39] who assumed spherical nanoparticles distributed homogeneously in the skutterudite matrix. This model includes an additional relaxation time written as

t np -1 = v s s -1 +s l -1 ( ) -1 d n (6)
where d n is the density of nanoparticles (in m -3 ) and  s and  l are the short-and long-wavelength scattering cross sections, respectively. This formula is based on a Matthiessen-like relation between the

k B v x = w / k B T L
short and long-wavelength diffusive regimes of phonons scattered by the nanoparticles. d n is related to the volume fraction and the radius of the nanoparticles by d n = f / (4R 3 /3). The two scattering cross sections are expressed as

s s = 2p R 2 (7)
s l = p R 2 4 9 DD D ae è ç ö ø ÷ 2 wR v ae è ç ö ø ÷ 4 ( 8
)
where D and are the difference in density between the matrix and the nanoparticles and the density of the matrix, respectively. The set of Eq.( 4), ( 5) and ( 6) contains no adjustable parameters so that the relaxation time of phonon diffusion by nanoparticles can be evaluated numerically as a function of both the volume fraction and the radius. We applied this approach to the In-filled and In-free nanocomposites (Refs. 25 and 27) to determine the role played by both a homogeneous distribution of nanoparticles and

In atoms on  l . As a first step, the lattice thermal conductivity data of the nanoparticle-free compounds were fitted according to Eq.( 4) with , , , and  0 as free parameters and using 2967 m.s -1 measured by a conventional pulse-echo method. The results obtained for CoSb 3 and In 0.2 Co 4 Sb 12 are listed in Table 1. Assuming that the presence of nanoparticles has a negligible influence on these values, we used them to compute the lattice thermal conductivity temperature dependences of the composites.

The influence of the nanoparticle sizes was first investigated keeping constant the volume fraction to 1.9 % (corresponding to 2 wt% of PbTe). Several interesting results emerge from these calculations compiled in Figure 1 for CoSb 3 and In 0.2 Co 4 Sb 12 containing either ZnO or PbTe nanoparticles. In the case of the parent compound CoSb 3 , an optimum size of the nanoparticles that minimizes exists (Figs 1a and1b). These sizes fall in the 6-8 and 2-4 nm ranges for PbTe and ZnO, respectively, resulting in respective decreases of the dielectric maximum of up to 66 and 73%. In addition, below 10 nm, the oxide nanoparticles have a stronger impact due to a more pronounced density difference. Upon

R D L A B C v = k l
introducing In in the skutterudite matrix, the optima sizes are shifted to 14 and 6 nm for PbTe and ZnO, respectively (Figs 1c and1d). This model further shows that nanoparticles primarily affect the lowtemperature region as it is seen from the strong decrease in the dielectric maximum of the k l (T ) curves.

At 300 K, their influence is lessened though still discernable. Yet, because this model is strictly applicable up to the Debye temperature (around 320 K in the present case), it is not possible to determine whether or not the observed effect would vanish at higher temperatures.

Figure 2 shows  l as a function of the nanoparticle size for the four series of composites at 300 K with a volume fraction of 1.9%. This plot clearly emphasizes the role played by the nanoparticle's size on the reduction of the lattice thermal conductivity. Surprisingly, the relative decrease in  l appears significantly more pronounced in In-filled samples than in CoSb 3 as shown in Figures 2b and2d. One may expect a weaker impact of the nanoparticles on  l due to the much lower values in the In-filled compounds with respect to CoSb 3 . This rather counterintuitive result may nevertheless be explained by the mass fluctuations induced by the In atoms. Meisner et al. [40] demonstrated that partially-filled skutterudites can be seen as a solid solution of unfilled and filled skutterudite phases. Because the longwavelength phonons are the main heat carriers in alloys, their mean free path exceeds the typical distance between nanoparticles. The probability for these phonons to be scattered will be therefore higher in partially filled compounds than in unfilled compounds thereby explaining the stronger reduction seen in In-filled compounds.

Finally, we adjusted the parameters of the model (volume fraction and grain size) to those of our PbTe/ZnO In 0.2 Co 4 Sb 12 samples to try to predict their thermal behavior. The results are plotted in Figures 2e and2f. From these plots, it is clear that the higher the volume fraction is, the higher the decrease in the lattice thermal conductivity. We also note that the dielectric maximum is shifted to higher temperatures. While for the PbTe nanocomposites the reduction amounts to 50% at 300 K, the reduction observed in the ZnO composites is lower (20%) even though still noticeable.

Phase and microstructure

For the PbTe-In 0.2 Co 4 Sb 12 series, the relative densities remain higher than 95%, except for the highest PbTe content (12%, see Table 2) for which it decreases to 90%. In the ZnO counterparts, the relative densities decrease down to 90% with increasing the ZnO content. For a similar weight ratio, the volume ratio will be higher for ZnO than for PbTe suggesting only little difference in density between the two series of samples. Surprisingly, the presence of indium seems to limit the densification retardation observed for the similarly prepared ZnO-CoSb 3 series studied in ref. [27,28]. Higher porosity induced by the presence of nanoparticles was already underlined in thermoelectric composites such as ZrO 2 /CoSb 3 [17], SiC/Bi 2 Te 3 [41] or ceramics [42]. Among the possible mechanisms that were invoked to explain this result [43], mechanical stresses induced by the presence of rigid inclusions seem to be more likely in the present case. However, the non-homogeneous distribution of the nanoparticles (see below) may also play a role.

The PXRD patterns of the different densified samples (Figure 3) show similar features. In addition to the main peaks that can be indexed to the cubic skutterudite structure, some additional peaks indicate the presence of a small amount of secondary phases (mainly InSb). The intensity of the diffraction peaks of PbTe (Figure 3a) and ZnO (Figure 3b) increases with their nominal fraction x, suggesting that the inclusions are not finely dispersed into the skutterudite matrix. This behaviour is similar to that observed in In-free composites PbTe-CoSb 3 and ZnO-CoSb 3 [25][26][27][28]. Interestingly, in the case of the 12wt%-ZnO specimen, a diffraction peak located around 29° was identified as being Zn 3 CoInSbO 8 (JCPDS: 04-008-3988), evidencing that a reaction occurred between the matrix and the nanoparticles during the SPS process. Shi et al. [44] have also observed the formation of a Ba 6 C 60 phase in their study on C 60 /Ba 0.44 Co 4 Sb 12 composites. However, the presence of this phase was found to strongly modify both the chemical composition and the unit cell parameter of the skutterudite structure.

In our case, no variations in the lattice parameter could be observed regardless of the ZnO mass fraction in the sample (Table 1). The same conclusion can be drawn for the PbTe series.

SEM images were collected in the backscattered electron mode (BSE) to determine the distribution of the phases in the different samples (Figure 4). PbTe agglomerates at grain boundaries (Figures 4b to 4d) as already seen in the PbTe-CoSb 3 series [25,26]. X-ray mapping (Figures 5a to 5f) further corroborates this conclusion and reveals that InSb is also located at the grain boundaries. Li et al.

[31] as well as Mallik et al. [33] have already observed that InSb phases are formed in indium filled skutterudites and located preferentially at the grain boundaries. The grain size of the skutterudite matrix is around 3-5 m, fairly lower than the grain size suggested by the location of either PbTe or InSb. The presence of ZnO in the skutterudite matrix results in an important loss of density that is obviously seen on the images (Figures 4e to 4h) as pores around the grains. Likewise the PbTe series, the ZnO particles are agglomerated at the grain boundaries and form large areas whose size can exceeds 50 µm (mottled black and white zones in Figure 4f). This kind of microstructures, where micron-sized clusters of nanoparticles with irregular shape surround the matrix grains, is not restricted to the present samples and was already observed in composites such as for example Yb 2 O 3 /Yb 0.25 Co 4 Sb 12 [16], ZrO 2 /ZrNiSn [45], C 60 /Bi 2 Te 3 [46], C 60 /R x Co 4 Sb 12 [47]. Observations on fractured samples revealed the tendency of the powder-like aspect of the oxide to agglomerate. The ultrasound mixing method employed here is thus not efficient enough to break and disperse the ZnO agglomerates contained in the commercial powder. As a result, this leads to the presence of very porous and large clusters of oxide that are not densified by the sintering temperature used (the sintering temperature of ZnO is near 1300 °C).

Electrical and galvanomagnetic properties

The temperature dependences of the thermopower S and the Hall coefficient R H are shown in In-free series, the influence of ZnO was attributed to the presence of potential barriers which no longer seem to be at play upon inserting In. This suggests that both the electron concentration and their scattering mechanisms should be independent with the ZnO concentration.

The electron concentrations n and Hall mobilities  H = R H / of the two series of composites are plotted as a function of temperature in Figures 7 and8. The room temperature values are listed in Table 1. Both series show similar variations in n with temperature. The values range from 1x10 20 up to 2x10 20 cm -3 at 300 K and decrease only slightly with temperature as expected in degenerate semiconductors.

This behavior is typical of n-type partially filled skutterudites [START_REF] Uher | Skutterudites: prospective novel thermoelectrics[END_REF]49]. Noteworthy, neither the presence of PbTe nor the presence of ZnO influences the electron concentration in agreement with the thermopower data. The Hall mobility of the In 0.2 Co 4 Sb 12 sample is in agreement with literature data [47,50]. The presence of PbTe, ZnO and pores does not modify the scattering mechanisms of electrons i.e. neutral impurities scattering below 80 K and acoustic phonons scattering above this temperature.

The influence of nanoparticles and pores is however detrimental to the  H values which decrease by a factor of 5 (1.4) at 5 K and 2 (1.2) at 300 K between the reference sample and the 12wt%-PbTe specimen (8wt%-ZnO).

Combining the galvanomagnetic and thermopower data, the reduced effective mass of electrons m*/m 0 (m 0 is the bare electron mass) were estimated at 300 K (Table 1) assuming a single-parabolicband model described in detail elsewhere [34]. For both series of composites, m* remain close to that of the reference sample, implying no obvious change in the electronic band structure. Figure 9 As can be seen, our results match the general trend of n-type partially-filled skutterudites. The slight increase in m*/m 0 with n might be related to the presence of several conduction bands at the  and H points of the Brillouin zone, a feature not included in the present model [34].

The temperature dependence of the electrical resistivity of the In partially-filled skutterudites follows an almost linear trend typical of degenerate semiconductors (Figure 10). If the insertion of In strongly reduce  with respect to CoSb 3 (by a factor of 6 at 300 K), the addition of either PbTe or ZnO has less incidence on the  (T) data (Figures 10a and10b). Yet,  drops sharply below 7.2 K in the PbTe-In 0.2 Co 4 Sb 12 composites (Figure 10a). This temperature corresponds to the superconducting transition temperature of Pb indicating that some PbTe nanoparticles decompose into elemental Pb and Te. While Pb does not substitute for Co nor for Sb, Te does substitute for Sb. This result further corroborates the trend observed in the S(T) data suggesting that a minute amount of Te substitute for Sb.

Between 7.2 and 50 K,  decreases slowly in the PbTe-based composites while the ZnO analogues shows a different behaviour (Figure 10b). Above 50 K, both series show trends similar to the reference In 0.2 Co 4 Sb 12 skutterudite, with a monotonically increase in  with temperature. The  values tend to increase with the amount of inclusions due to the observed degradation of the Hall carrier mobility, a behavior well documented for various nanocomposite systems [17,[START_REF] Da Ros | Proc. Mater. Res. Soc. Symp[END_REF][57][58][59][60][61].

Thermal transport

The temperature dependence of the total thermal conductivity for all samples are depicted in Figures 11a and11b. Near room temperature some deviations between the low and high temperature measurements are evidenced. This is attributed to the different methods used to probe the thermal conductivity. In the low temperature range, thermal losses by radiation are present. They start to play a role for T > 150 K and increase with T according to a T 3 law. As a result, the thermal conductivity is overestimated when 150 < T ≤ 300 K. The electronic contribution  e was estimated using the Wiedemann-Franz law ( e = LT/ where L is the Lorenz number taken to the value of a degenerate electron gas L 0 = 2.44×10 -8 V 2 K -2 ). The ambipolar contribution dominating above 600 K was not taken into account in the following analysis. The lattice thermal conductivity  l was then deduced from  =  e +  l . The two contributions κ e and κ l are plotted in Figures 11c and11d for the PbTe and ZnO series, respectively. In In 0.2 Co 4 Sb 12 , the dielectric maximum is divided by eight with respect to CoSb 3 and the room-temperature value is reduced by 70%. This two main features originate from the presence of In in the cage-like structure of CoSb 3 . The dispersion of ZnO leads to a slight decrease in κ l below 300 K with the amount of nanoparticles up to 8%. The increase observed for 12% is most likely due to the lower density of this sample. Above 300 K, the presence of nanoparticles has only little influence leading to quasi-constant κ l values. No clear correlation between the fraction of PbTe nanoparticles and κ l is evidenced by our data. This is presumably due to varying amount of Te substituting for Sb in the skutterudite matrix. The exact influence of the nanoparticles is nevertheless difficult to unveil since In, the porosity and the nanoparticles that tend to agglomerate play a role in determining κ l . To lessen the impact of the former, we corrected the total  values following Eq.( 4). [START_REF] Landauer | Electrical conductivity in inhomogeneous media[END_REF] (9)

The results are reported in Figure 12 for both series of samples, showing only a weak decrease in the total thermal conductivity in fully dense samples. Because of the tendency of the nanoparticles to agglomerate, the comparison of the theoretical model with our experimental data in the 5 -300 K temperature range is quite delicate. According to the model (see Figure 2), increasing the diameter of the particles decreases the corresponding scattering rate of phonons. For diameters higher than 300 nm, only a modest reduction in the lattice thermal conductivity may be expected. This expectation is in qualitatively good agreement with the results shown in Figure 12. Nevertheless, it appears difficult to draw a final conclusion due to the complex microstructure of our samples that inevitably plays a role in determining the thermal transport.

Dimensionless thermoelectric figure of merit ZT

The calculated dimensionless figure of merit ZT of the different composites are presented in Figure 13 as a function of temperature. Compared to CoSb 3 , the introduction of In leads to a maximum ZT of 1.0 at 700 K, due to higher thermopower and lower thermal conductivity values. Both series of composites exhibit temperature dependences quite similar to the reference skutterudite. The values are however systematically lower in the whole temperature range. An exception to this trend is observed in the 2wt%-PbTe and 2wt%-ZnO samples that exhibit similar values up to 500 K. These results differ from those obtained in the ZnO-CoSb 3 series for which, a slight increase in ZT was observed at low ZnO contents [27]. At higher temperatures, the ZT of the PbTe series seems more detrimentally affected than that of the ZnO series, perhaps due to the minute amount of Te substituting for Sb as previously observed in the PbTe-CoSb 3 series [25]. Nevertheless, the ZT values of the ZnO series range between 0.78 and 1.03 at 700 K.

1 3 2 % 100   d  

Conclusion

We presented the transport properties of PbTe-In 0.2 Co 4 Sb 12 and ZnO-In 0.2 Co 4 Sb 12 nanocomposites between 2 and 800 K. SEM experiments revealed the tendency of the nanoparticles to agglomerate at the grain boundaries by forming large clusters. As a result, this non-homogeneously dispersion leads to a loss of density. The transport properties of both series of sample show similar temperature dependences suggesting that the nature of the nanoparticles (insulating versus semiconducting) does not play a significant role in the present case. The n-type electrical conduction of the parent In 0.2 Co 4 Sb 12 skutterudite is preserved upon introducing nanoparticles. While this conclusion holds true for the scattering mechanisms of electrons, this is no longer the case for the electron mobilities that tend to decrease with increasing the nanoparticle content. The analysis of the thermal properties revealed to be quite complex because of the presence of In, porosity and nano-inclusions that may all be a diffusion source of phonons. Generally, as expected, a reduction in the thermal conductivity was observed for both types of nanoparticles. The main aspects of the influence of nanoparticles on k l was captured by a model that explicitly takes into account phonon scattering by spherically-shaped particles. Despite this reduction and contrarily to results observed in e.g. Ba-filled skutterrudites with Ag nanoparticles [24], the high ZT values of the In 0.2 Co 4 Sb 12 matrix (ZT > 1) is not improved by the presence of nanoinclusions.

  ) which takes into account the theoretical densities (d i ) and the mass fraction (x) of each species (d CoSb3 = 7.64; d ZnO = 5.67; d PbTe = 8.27):

Figure 6

 6 Figure 6 and in the inset, respectively. Both S and R H are negative over the entire temperature range

  displays a comparison of the evolution of m*/m 0 as a function of the carrier concentration n in our materials and for n-type skutterudites R x Co 4 Sb 12 (R = Ba[51,52] -Tl[53] -Nd[54] -Ce[55] -La[56] -Ca[START_REF] Uher | Recent advances in the development of efficient n-type skutteudites[END_REF]49]).

Figure caption Figure 1 :

 caption1 Figure caption

Figure 2 :

 2 Figure 2: Theoretical influence of the PbTe or ZnO particles size on the reduction in the lattice thermal

Figure 3 :

 3 Figure 3: XRD patterns of the PbTe-(a) and ZnO-In 0.2 Co 4 Sb 12 (b) composites. Note the minute amount

Figure 4 :

 4 Figure 4: SEM backscattered electron images of a) In 0.2 Co 4 Sb 12 , b-d) PbTe-In 0.2 Co 4 Sb 12 composites and

Figure 5 :

 5 Figure 5: BSE images and their corresponding X-ray mapping images for a)-f) 8wt%PbTe-In 0.2 Co 4 Sb 12

Figure 6 :

 6 Figure 6: Temperature dependences of the thermopower S for a) PbTe-In 0.2 Co 4 Sb 12 composites and b)

Figure 7 :

 7 Figure 7: Temperature dependences of the carrier concentrations n for a) PbTe-In 0.2 Co 4 Sb 12 composites

Figure 8 :

 8 Figure 8: Temperature dependences of the Hall mobility  H for a) PbTe-In 0.2 Co 4 Sb 12 composites and b)

Figure 9 :

 9 Figure 9: Evolution of the reduced effective mass m*/m 0 as a function of the carrier concentration for

Figure 10 :

 10 Figure 10: Temperature dependences of the resistivity  for a) PbTe-In 0.2 Co 4 Sb 12 composites and b)

Figure 11 :

 11 Figure 11: Temperature dependences of the total thermal conductivity, , for a) PbTe-In 0.2 Co 4 Sb 12 composites and b) ZnO-In 0.2 Co 4 Sb 12 composites. Temperature dependences of the electronic  (solid lines) and lattice  (symbols) thermal contributions for c) PbTe-In 0.2 Co 4 Sb 12 composites and d) ZnO-

Figure 12 :

 12 Figure 12: Temperature dependences of the total thermal conductivity corrected from porosity

Figure 13 :

 13 Figure 13: Temperature dependences of the dimensionless figure of merit ZT for a) PbTe-In 0.2 Co 4 Sb 12

  Figure

Figure

  Figure

Table 1 :

 1 Values of the fitting parameters used to describe the lattice thermal conductivity of the CoSb 3 and In 0.2 Co 4 Sb 12 skutterudites.

	Material	L (μm)	A (10 -43 s 3 )	B (10 -18 s.K -1 )	C (10 33 s -3 )	 0 (THz)
	Co 4 Sb 12	4.4	3.4	3.3	/	/
	In 0.2 Co 4 Sb 12	4.9	137.7	2.8	2.6	6.2

Table 2 :

 2 Theoretical density d th , relative density d rel , and cell parameter a (evaluated by Rietveld refinements of the PXRD diagrams) of the PbTe-In 0.2 Co 4 Sb 12 and ZnO-In 0.2 Co 4 Sb 12 composites. The carrier concentrations n, Hall mobilities µ H , thermopower S were measured at 300 K. The reduced effective masses of electrons m*/m 0 were estimated at 300 K.

	Composite	d th	d rel (%)	a (Å)	n (cm -3 )	µ H (cm 2 V -1 s -1 )	S (µVK -1 )	m*/m 0
	In 0.2 Co 4 Sb 12 7.69	98	9.0565(2)	1.87x10 20	49	-145	2.6
	+ 2% PbTe	7.65	96	9.0563(2)	1.86x10 20	37	-159	3.0
	+ 8% PbTe	7.69	96	9.0559(2)	1.19x10 20	48	-162	2.3
	+ 12% PbTe	7.71	90	9.0565(2)	1.35x10 20	29	-162	2.5
	+ 2% ZnO	7.59	98	9.0567(2)	1.80x10 20	49	-143	2.6
	+ 4% ZnO	7.54	94	9.0562(2)	1.32x10 20	51	-144	2.1
	+ 8% ZnO	7.43	89	9.0565(2)	1.10x10 20	41	-143	1.8
	+ 12% ZnO	7.33	89	9.0564(2)	1.54x10 20	34	-146	2.3

  Highlights Fabrication of nanostructured skutterudites ZnO or PbTe/In 0.2 Co 4 Sb 12  Thermal conductivity modeling accounts for experimental results  Greater lattice thermal conductivity decrease in In 0.2 Co 4 Sb 12 than in CoSb 3

	*Highlights (for review)
		50								CoSb	3
										2 nm
		40								4 nm
	)									6 nm
	K -1									8 nm
	(Wm -1	30								12 nm 10 nm
	l
	!	20								14 nm
										16 nm
		10		18 nm  A max ZT of 1.05 is obtained at 700 K in a 2wt.% ZnO-containing sample 20 nm
										100 nm
		0	0	50	100	150	200	250	300	1000 nm
						T (K)

Acknowledgments

The authors acknowledge support from the European Network of Excellence CMA 'Complex Metallic Alloys'.


Axel Hutt
email: axel.hutt@loria.fr

Connie Sutherland

André Longtin

Driving neural oscillations with correlated spatial input and topographic feedback

Keywords: number͑s͒: 87.19.lp, 02.30.Ks, 02.50.Ey, 87.19.lr

We consider how oscillatory activity in networks of excitable systems depends on spatial correlations of random inputs and the spatial range of feedback coupling. Analysis of a neural field model with topographic delayed recurrent feedback reveals how oscillations in certain frequency bands, including the gamma band, are enhanced by increases in the input correlation length. Further, the enhancement is maximal when this length exceeds the feedback coupling range. Suppression of oscillatory power occurs concomitantly in other bands. These effects depend solely on the ratio of input and feedback length scales. The precise positions of these bands are determined by the synaptic constants and the delays. The results agree with numerical simulations of the model and of a network of stochastic spiking neurons, and are expected for any noise-driven excitable element networks.

I. INTRODUCTION

There has been much interest recently in the dynamics of nonlinear systems subjected to spatiotemporal forcing ͓1͔. The interaction of the characteristic length and time scales of the input with those of the system is important for understanding the kinds of patterns that can occur under such forcing, and is still an open question despite significant progress. In some contexts, such as in neural or metabolic networks, this forcing contains information that the system must adequately respond to, often via feedback ͓2͔. Sensory systems, for example, are organized in a feedforward manner, where stimulus information is relayed from primary sensory neurons to higher brain centers, and these in turn feed back to primary areas. This enables patterned responses to specific stimulus features, which the feedback can reinforce or modulate. Of key interest is the induction of patterns in space such as localized bumps ͓3͔ or temporal oscillatory behavior ͓4,5͔. There is evidence that oscillations can be important for biasing input selection or enhancing selectivity to stimulus motion ͓6͔, altering the connection strength between neurons, and temporarily linking neurons to assemblies ͓7͔.

In its simplest form, the receptive field of a given neuron is defined by the feedforward inputs its receives, and thus is made up of all the neurons ͑and their associated stimulus space͒ that have an effect on its firing activity. This effect can be excitatory as in the classic center receptive field, or inhibitory, as in the classic inhibitory surround. However, this picture is complicated by feedback pathways in different forms. For example, in the visual system of macaque monkeys, the retina projects in a topographic feedforward manner to the lateral geniculate ͑LGN͒, which in turn projects to the primary visual cortex V1. But there are also lateral ͑"horizontal"͒ connections within V1. And V1 projects to other areas, which in turn project back to V1 ͑see, e.g., ͓8͔͒. These feedback connections shape the classic receptive field and its dependence on stimulus features ͑such as spatial fre-quency, orientation tuning, and stimulus contrast͒, and also mediate long-range effects that modulate the classic receptive field ͓9,10͔. There is debate, for example, as to whether the receptive fields of cells in V1 are significantly determined by fast ͑short-delay͒ feedback projections to V1 in conjunction with horizontal connections within V1, rather than by feedforward projection from LGN and horizontal connections within V1. A recent modeling study in fact finds that feedback from higher areas to V1 is a very important determinant of receptive field structure of V1 neurons, and can mediate inhibitory influences in a nontrivial way ͓11͔.

Thus the existence of feedback begs an extension of the notion of receptive field or matched filter commonly used to understand how spatial inputs are processed in feedforward networks ͓12͔. Our work considers how such spatially organized feedback affects neural firing activity using an effective receptive field formulation, and focuses closely on oscillatory activity rather than on the genesis of detailed receptive field structure in any particular system. To this end we consider below a system where feedback projections are strong and horizontal connections weak, and are known to mediate oscillations that depend on the spatial structure of the input.

Neural feedback can act globally, or be organized topographically, i.e., locally ͓13,14͔ ͑see Fig. 1͒ such that a neighborhood of cells projects back near itself via feedback, often after a significant delay. There has been much work on the patterns of activity in recurrent networks ͑see, e.g., ͓3,13,15,16͔͒, but little work on their response to spatiotemporal inputs. It is known theoretically that oscillations can be induced in such networks with global delayed feedback when fluctuating inputs to the cells are spatially uniform ͓17-20͔, or where each cell receives an adjustable mixture of individual noise and common ͑global͒ noise, in agreement with experiment ͓4,17,21͔. These studies focused on global feedback pathways, and thus lack a notion of "neighborhood:" the connections between cells are via global feedback, and the input to each cell is either global or a mixture of global and local noise, i.e., it lacks a clearly defined characteristic length scale. However, little is known about the dynamic role played by topographic feedback pathways which are also known to exist. In particular, their existence raises the question of how the length scales of feedback interact with the spatial scale of time-varying forcing signals to correlate activity and produce oscillations in excitable systems.

In this work we introduce a model of two neural populations interacting nonlocally via excitatory and inhibitory delayed feedback connections. Thus there are no local horizontal connections within the populations. Further, the topography of feedback connections is studied over a wide range of spatial scales; at the extreme "local" feedback case a neuron essentially feeds back to itself via the other population, while at the global case each neuron affects every other neuron via feedback, as in previous studies ͓4,17,21͔. Moreover, we look at how such local and global delayed feedbacks interact with spatiotemporal noise based on a population rate formulation of the problem, and compare theory to numerics on this model as well as on a realistic network of stochastic spiking neurons.

II. THE POPULATION RATE MODEL

The model we investigate is grounded in the electrosensory system of electric fish ͑for anatomical reviews, see ͓22͔͒, but similar configurations can also be found in parts of the vertebrate brain ͓23-25͔, as in many other nonlinear spatial systems. It is made up of two neural populations: ͑1͒ the ELL, a layer of pyramidal cells driven by the primary receptors that receive the stimulus, and ͑2͒ the higher area Np. These are spatially coupled via feedback, according to normalized connectivity kernels K en ͑x͒ and K ne ͑x͒ denoting connections from the Np͑n͒ to the ELL͑e͒ and vice versa, respectively ͑see Fig. 1͒. In contrast to these interarea connections, the neurons in both populations have insignificant direct coupling ͑i.e., horizontal connections͒, which reflects anatomical findings ͓22͔. Further, the coupling from the ELL to Np is excitatory and delayed in time by 1 , while the coupling back to the ELL is inhibitory ͓22͔ with delay 2 . Moreover, the model considers excitatory spatiotemporal stimuli I͑x , t͒, which for simplicity are directly relayed to ELL by the receptors.

The present work introduces a model which describes mathematically the mean membrane potential in neural populations ͓26,27͔ and which are strongly related to experimentally observable local field potentials ͓28,29͔. In this description, membrane potentials and firing rates are temporal mean values over a time interval smaller than the typical oscillation period of the systems activity. We focus on how the spatial correlations in the input stimulus interact with those imposed by the topographic feedback to produce network oscillations. Specifically, we will consider below a spatiotemporal noise with a well-defined and adjustable spatial correlation length. We focus on a general scenario where two neural populations interact via feedback; one of them is driven by the spatiotemporal noise, which represents input from another part of the brain, or from a physical stimulus external to the animal ͑perhaps relayed via primary recep-tors͒. We will see how this general case can be replaced by a simpler circuit with self-feedback and a compound connectivity and delay.

The evolution equations read

in ץE͑x,t͒ ץt = -E͑x,t͒ + I͑x,t͒ -g in͵ ⍀ n dy K en ͑x -y͒S n ͓N͑y,t -2 ͔͒, ͑1͒ ex ץN͑y,t͒ ץt = -N͑y,t͒ + g ex͵ ⍀ e dx K ne ͑y -x͒S e ͓E͑x,t -1 ͔͒, ͑2͒
where ⍀ e and ⍀ n are the respective domains of the ELL and the Np; we further assume periodic boundary conditions. Further E , N and S e , S n denote the mean membrane potential and mean firing rate of principal neurons in ELL and Np, respectively. In addition, the afferent fibers to the Np terminate at excitatory synapses with synaptic efficacy g ex and response time constant ex , while the inhibitory synapses of the afferents to the ELL have the synaptic efficacy g in and the response time constant in .

To examine the strength of oscillatory activity in the ELL, we assume spatially homogeneous stationary states E 0 = const, N 0 = const in the absence of external stimuli ͓I͑x , t͒ =0͔ and study small deviations from these states. For sigmoidal firing rate functions, we find a single stationary state that is a solution of E 0 =-g in S n ͓g ex S e ͑E 0 ͔͒, N 0 = g ex S e ͑E 0 ͒. This stationary state is interpreted as the resting state of the system, which is perturbed by incoming external inputs. After linearizing Eqs. ͑1͒ and ͑2͒, and reintroducing the small input I͑x , t͒ with vanishing spatial and temporal mean, i.e., ͗I͑x , t͒͘ = 0, we find that the membrane potential deviations in the ELL, u͑x , t͒ = E͑x , t͒ -E 0 , obey

L ˆu͑x,t͒ = - g ex in ͵ ⍀ e dxЈF͑x -xЈ͒u͑xЈ,t -d ͒ + I͑x,t͒

͑3͒

with the nonlinear gain g ϵ͑ץS e / ץE͒͑ץS n / ץN͒g ex g in evaluated at the resting state ͑E 0 , N 0 ͒, the sum of the time delays d = 1 + 2 , and the temporal operator

L ˆ= ץ 2 / ץt 2 + ͑1 / ex _ τ 2 τ 1 K en (x) K ne (x) I(x,t) Ell Np + FIG. 1.
Topography of the feedback model under study. The plus and minus signs indicate the excitatory and inhibitory connections, respectively. The bell-shaped distributions K en ͑x͒ and K ne ͑x͒ represent the spatial connectivity functions, while the distribution I͑x , t͒ denotes the spatial correlation function of the external input.

+1/ in ץ͒ / ץt +1/ ex in . Moreover, we find that the system exhibits an effective spatial feedback kernel

F͑x -xЈ͒ = ͵ ⍀ n dy K en ͑x -y͒K ne ͑y -xЈ͒. ͑4͒
In other words, the two spatial interactions collapse to a single spatial interaction and the effective spatial feedback kernel is the convolution of the kernels to and from the area Np. This kernel, referred to below as the feedback kernel, can be interpreted as a lateral connectivity function of the ELL cells, keeping in mind that these connections are made via another nucleus, and thus involve delays. In the following, the feedback kernel is chosen as the Gaussian distribution F͑z͒ = exp͑-z 2 / 2 f 2 ͒ / ͱ 2 f with the standard deviation f representing the spatial extent of the feedback. This choice preserves the total synaptic strength on each neuron irrespective of f . Moreover, the stability of the resting state is given by the complex roots C of the characteristic equation ex in 2 + ͑ ex + in ͒ +1+gF ˜͑k͒exp͑d ͒ = 0, where F ˜͑k͒ is the spatial Fourier transform of the feedback kernel F and k denotes the wave number. The root is the Lyapunov exponent of ͑3͒ and thus the resting state is linearly stable if the real part of is negative, i.e., R͑͒ Ͻ 0. In the following we do not study the linear stability in detail but use parameters which guarantee the stability of the resting state. Now we assume the external input I͑x , t͒ as a small external perturbation, which maintains the system in the linear regime about the resting state. In the following, we choose I͑x , t͒ as spatiotemporal noise with ͗I͑x,t 1 ͒I͑y,t 2 ͒͘ = Q␦͑t 1t 2 ͒C͑x -y͒, ͑5͒

with C͑z͒ = exp͑-z 2 / 2 i 2 ͒ / ͱ 2 and where Q is the noise vari- ance. This means the external stimulus is uncorrelated in time ͑"white"͒ and Gaussian correlated in space with correlation length i . Moreover, the intensity of the external input ͗I 2 ͑x , t͒͘ is a constant independent of i .

III. THE POWER SPECTRUM

To compute the power spectrum of u͑x , t͒, we introduce the Green's function G͑x , t͒ by

u͑x,t͒ = ͵ ⍀ e dz ͵ -ϱ ϱ dT G͑x -z,t -T͒I͑z,T͒.
Inserting this ansatz into Eq. ͑3͒ we gain the expression

L ˆG͑x,t͒ + s ͵ ⍀ e dxЈF͑xЈ͒u͑x -xЈ,t -d ͒ = ␦͑x,t -d ͒
with s = g / ex in . The subsequent Fourier transform into the k-space yields the Fourier transform of G͑x , t͒ and, consequently, by the back transformation into x-t space one obtains

G͑x,t͒ = 1 ͑2͒ 2 ͵ -ϱ ϱ dk ͵ -ϱ ϱ d ϫ e ikx-i͑t-͒ L͑-i͒ + s ͱ 2F ˜͑k͒exp͑i͒ .
Then the autocorrelation function ͗u * ͑x , t 1 ͒u͑y , t 2 ͒͘ of the resulting membrane potential can be computed by applying the input correlation function ͑5͒ ͓30͔. Here and in the following, the superscript ء denotes the complex conjugate. By virtue of the spatial homogeneity and the temporal stationarity, we find ͗u * ͑x , t 1 ͒u͑y ,

t 2 ͒͘ = K͑x -y , t 1 -t 2 ͒ with K͑x,t͒ = Q ͱ 2 ͵ -ϱ ϱ d ͵ -ϱ ϱ dk C ¯͑k͉͒G ˜͑k,͉͒ 2 e it-ikx
and the Fourier transform of the input correlation function C ¯͑k͒. This relation allows us to apply the Wiener-Khinchin theorem ͓31͔, i.e., the temporal power spectrum of u͑x , t͒ is the Fourier transform of the autocorrelation function K͑0,t͒.

Consequently we obtain the power spectrum of the membrane potential in the ELL,

P͑,͒ = Q ͵ -ϱ ϱ dl C ˜͑l͒ A͑͒ + B͑͒F ˜͑l͒ + DF ˜2͑l͒

͑6͒

with the frequency , the function A͑͒, the constant D, and B͑͒ =2s ͱ 2͓cos͑2͒͑␣␤ -4 2 2 ͒ -2 sin͑2͒͑␣ + ␤͔͒.

In addition we rescaled space by x → x / i and thus l = i k represents the dimensionless wave number. Consequently the Fourier transforms of the feedback kernel F͑l͒ and the input correlation function C͑l͒ in dimensionless wave numbers read

C ˜͑l͒ = exp͑-l 2 /2͒, F ˜͑l͒ = exp͑-2 l 2 /2͒.

͑7͒

In addition we have introduced as the ratio of the two characteristic length scales that correlate neural activity in the problem, namely, the feedback spread and the input correlation length, i.e., = f / i . Consequently the relation of the feedback and the input correlation ranges defines the power spectrum in Eq. ͑6͒.

To compute the power spectrum numerically, we expanded u͑x , t͒ into a discrete Fourier basis where the stochastic integral-differential equation ͑3͒ is a set of uncoupled stochastic delay differential equations ͑SDDEs͒. The spatially correlated input was implemented according to the rules given in ͓32͔. Since each SDDE gives the time evolution of a spatial mode, we integrated each equation separately, Fourier recomposed the obtained solutions to gain u͑x , t͒, and computed the temporal power spectrum by a temporal Fourier transform. Figure 2 shows the power spectrum in the ELL for a large, medium, and low value of . We observe a clear power maximum at about 40 Hz for =1/ 40, i.e., for large input correlations compared to the feedback range, while the larger value = 1 yields a peak at the same frequency with lower power. In contrast, a large value = 40 makes the peak at 40 Hz vanish and leads to a power peak at 0 Hz.

To better understand the power peak and its dependence on the relation of feedback and input correlation ranges, we examine the sign of dP͑ , ͒ / d for different frequencies and a constant feedback range f . Taking a closer look at the power spectrum in Eq. ͑6͒, occurs in F ˜͑l͒ only and we find

ץP͑,͒ ץ

= Q ͵ -ϱ ϱ dl 2l 2 2 C ˜͑l͒F ˜͑l͒ ͓A͑͒ + B͑͒F ˜͑l͒ + DF ˜2͑l͔͒ 2 Ͼ0 ϫ ͓B͑͒ + 2DF ˜͑l͔͒. ͑8͒
Since the last term in ͑8͒ determines the sign of dP͑ , ͒ / d, one can distinguish two cases. ͑1͒ In one case there are some frequencies for which dP͑ , ͒ / d Ͻ 0, i.e., the power decreases with increasing and thus the power increases with increasing input correlation. Equation ͑8͒ reveals that this case occurs only if B͑͒ Ͻ 0 ͑necessary condition͒, i.e., with frequencies in the intervals

m Ͻ Ͻ m+1 with tan͑2 j d ͒ = 1 -␣ j 2 ␤ j ͑9͒
and m Z 0 , m Ͻ ͑1+2m͒ / 4 d Ͻ m+1 , ␣ =4 2 ex in , and ␤ = ͑ ex + in ͒2. In other words, high ͑low͒ power for large ͑small͒ input correlations is present in frequency bands ͓ 0 ; 1 ͔ , ͓ 2 ; 3 ͔ ,... around 1 / 4 d ,5/ 4 d ,..., and the bandwidths are determined by the total delay time d and the synaptic response time constants ex and in . ͑2͒ In the other case dP͑ , ͒ / d Ͼ 0, there are some frequencies at which the power increases with increasing , i.e., the power increases with decreasing input correlation. From ͑8͒ it is obvious that this case occurs for B͑͒ Ͼ 0 ͑sufficient condition͒ in the complementary frequency bands ͓0; 0 ͔ , ͓ 1 ; 2 ͔ ,.... Figure 3 shows the intervals of the necessary condition ͑9͒ ͑dashed lines͒ and the exact bands computed numerically from Eq. ͑6͒ for different values of . First we observe that the frequency bands defined by ͑9͒ are constant with respect to . They are excellent estimates of the real bands determined numerically from ͑6͒, while slightly changes the numerical borders of the bands. Further, the frequency bands with dP / d Ͻ 0 ͑dP / d Ͼ 0͒ at about 40 ͑0͒ Hz show good accordance with the band of frequencies in Fig. 2 which exhibits large ͑small͒ power for low values of .

Consequently, the power peak at nonzero frequency in Fig. 2 is defined by the total feedback delay d , and the frequency bands of enhanced and attenuated power mainly depend on the feedback delay and the synaptic time constants in the feedback system. Hence the observed frequency bands are rather independent of the external input, as expected from a linearized neural field theory. In contrast, the external input determines the enhancement and attenuation of the oscillations in the frequency bands.

To gain further insight into this mechanism, we can consider the contributions of different factors to the integral that defines the oscillatory power in the time domain P͑ , ͒. Recall the form of the power spectrum P͑ , ͒ = ͐ -ϱ ϱ dl R͑ , l͒C ˜͑l͒ in Eq. ͑6͒.

Here R͑ , l͒ represents the response function of the system. The total power thus involves multiplication of the response by the spatial Fourier transform of the noise, followed by an integral over all wave numbers. Figure 4 shows R͑ , l͒ and R͑ , l͒C ˜͑l͒ for two values of in the wave number range 0 ഛ l ഛ 4. For the sake of discussion, we can assume that f was changed, while i remained constant.

In the case of =1/ 40 ͓Fig. 4͑a͔͒, we find F ˜͑l͒Ϸ1 for this range of wave numbers ͑not shown͒. Consequently, we observe a strong system response in the temporal frequency range around Ϸ 40 Hz, and lower responses away from this range, for all wave numbers shown. Considering the product of R͑ , l͒ and the fast-decaying spatial correlation function C ˜͑l͒ = exp͑-l 2 / 2͒, which is independent of the frequency ͓Fig. 4͑b͔͒, after integration over all values of l we find the resulting power peak at about 40 Hz ͑see Fig. 2͒.

In contrast, = 40 yields a rapidly decreasing function F ˜͑l͒ with increasing l and a strong response at low frequencies for all wave numbers, while R͑ , l͒ exhibits similar val- ues at ϳ40 Hz for low wave numbers only ͓Fig. 4͑c͔͒. Consequently the product R͑ , l͒C ˜͑l͒ exhibits large values at low frequencies for high wave numbers while R͑ , l͒C ˜͑l͒ shows similar values at ϳ40 Hz for very small l only ͓Fig. 4͑c͔͒. Hence integrating R͑ , l͒C ˜͑l͒ over all l yields high power values at low frequencies only, outside the gamma range ͑see Fig. 2͒. So we see that the spatiotemporal structure of the feedback system sets up temporal resonances that depend on the wave number. The parameter determines whether there is one or more resonance along the frequency axis ͓compare Figs. 4͑a͒ and 4͑c͔͒ as well as the range of wave numbers over which the response is relatively strong. A temporal resonance will thus be visible in the time evolution of the system, provided the input noise has sufficient spectral power at lower wave numbers in order to produce a sizable integrand near that resonance. High power at such low wave numbers ͑longer wavelengths͒ is seen when the noise is strongly correlated in space-and the difference between "long" and "short" is set by the feedback range, i.e., it is the value of that counts. The result is that, for a given feedback range, the total power shifts from zero frequency toward frequencies in the gamma range as decreases. One can see intuitively that, even in the absence of input, the topographic feedback reinforces firing rate oscillations in local neighborhoods because the feedback pulls such neighborhoods together. In the presence of input, local oscillations are further reinforced when a local neighborhood is driven by a coherent noise, which occurs when the input correlation length exceeds the feedback range of connectivity ͑ i Ͼ f ͒. The frequency band around 40 Hz where power increases with is one that agrees with experimental observations where input correlation produces gamma oscillations in the context of global feedback ͓4͔. The actual frequency range where this effect will be seen depends on the specific parameters of the problem, but qualitatively, it will always be seen. What our results further indicate is that this increase can also occur if the feedback is local rather than global. In this case less input correlation is required to see increased gamma power.

This raises the interesting possibility that a sensory system may detect changes in gamma strength as a function of input parameters, i.e., to signal features of the input such as spatial correlation via strength in the gamma band ͓33͔. One can then ask how much input correlation affects the gamma power. The answer, interestingly, depends only on f and i via their ratio, , as Fig. 5 reveals. This figure first shows in its left panels that increasing the correlation of the input, i.e., increasing 1 / , produces a peak in the gamma range, as in Fig. 2.

Figure 5͑c͒ further plots the integrated power ⌫ in the gamma range for many different combinations of f and i . Note that changes in f are done by conserving the total synaptic strength onto a given neuron. Further, changes in i are done at constant total stimulus power. When i is small in comparison with the feedback spread, the oscillation is weak. This can change if i increases, f decreases, or both. Opposite behavior occurs in the complementary bands. All results strikingly fall on the same curve predicted by the theory. Significant oscillations arise when i approaches or exceeds f , i.e., when is ϳ1 or less.

IV. THE SPIKING NEURON MODEL

We finally verified our results against simulations of a network of N stochastic leaky integrate-and-fire ͑LIF͒ neurons with delayed spike coupling, as in ͓17͔, except that the feedback is no longer global: there is a parameter that specifies the spatial spread of the feedback. Each LIF neuron evolves according to

dV j dt = -V j + + j ͑t͒ + I j ͑t͒ + g N ͵ d ϱ dtЈh͑tЈ -d ͒ ͚ k=1 N F͑j -k͒x k ͑t -tЈ͒, ͑10͒
where V j represents the membrane potential of the jth neuron in the network, is the bias ͑the same for all cells͒, and each cell receives a time-varying input noise I j ͑t͒ taken from Eq. ͑5͒ with correlation function C͑j͒. Further, each neuron has its own internal noise j ͑t͒ modeled by Gaussian white noise of intensity D, exhibits a refractory period r , and responds to incoming feedback activity by h͑t͒ = ␣ 2 ͑t͒exp͑-␣t͒ with the feedback kernel F of the neural field model. for different combinations of f and i the gamma power again falls on a single curve, with a similar qualitative feature as for the neural field model.

V. CONCLUSION

In summary, our work describes the influence of a spatiotemporal noise with a well-defined length scale on a delayed negative feedback network. We have shown using theory and numerics that increasing input correlation length can enhance or suppress oscillations in topographic feedback networks with delay. This continues until saturation when the input correlation length i exceeds f , i.e., the effective spatial spread of both feedforward and feedback connections. The topographic feedback enables the system to gauge the spatial correlation of the input by the power in the gamma range. This occurs when the input spatial mode amplitudes are strong and their temporal part is in the gamma range, a condition met when i ജ f . This also requires a stimulus temporal bandwidth that encompasses the resonance in the gamma range. We point out that this behavior is a property of both neural field models and spiking networks of stochastic neurons. The former are related to local field potentials ͑LFPs͒, while the latter are related to spiking activity, and the equivalence between these two kinds of measurements for various phenomena is a matter of ongoing debate. Recent work reports coherence between LFPs and spiking activity in visual cortex ͓5͔ and parietal cortex ͓34͔, and our results predict another instance where this equivalence might be observed experimentally.

The synergy between the correlations imposed by spatiotemporal input and those imposed by feedback to produce oscillatory firing activity are general properties of topographic networks, as is the antisynergy in other frequency bands. This complements findings in systems with lateral connections ͓14,35͔ where matching input correlation to lateral connectivity causes the highest firing rates. Our results can be generalized to multiple feedback loops, both positive and negative, and can also incorporate the effect of local connectivity, as well as distributions of delays rather than fixed delays. 1/η=σ i /σ f 1/η=σ i /σ f 1/η=100 1/η=0

x FIG. 5. Comparison of spectral power in neural fields and spiking neuron networks. ͑a͒ Power spectra of the neural field for a small ͑1 / =1/ 40͒ and large ͑1 / =40͒ stimulus correlation length relative to the feedback spread. ͑b͒ Power spectra of the spike train from a single stochastic neuron embedded in a network of 100 spiking neurons with topographic delayed feedback for small ͑1 / =0͒ and large ͑ = 100͒ stimulus correlation length. Each neuron is governed by Eq. ͑10͒. ͑c͒ Integrated power ⌫ in the gamma range ͑30, 50͒ Hz as a function of 1 / for the neural field model ͓Eq. ͑6͔͒. ͑d͒ Same as ͑c͒ except for the spiking neuron network with the power measured in the range ͑20, 40͒ Hz. The parameters of ͑a͒ and ͑c͒ are the same as in Fig. 2. The parameters in ͑b͒ and ͑d͒ are = 0.5 ͑subthreshold regime͒, g = -0.6, D = 0.08, ␣ =3, d =1, r = 0.1, the spiking threshold is 1, the voltage reset is 0, and all times are in units of the membrane constant, assumed to be 6 ms.

FIG. 2 .FIG. 3 .

 23 FIG. 2.Power spectrum of the membrane potential E about the stationary state E 0 for different ratios of the feedback and the input correlation range. For the numerical simulations, a temporal time step of 0.1 ms and a spatial discretization step of 5 m with 10 5 grid points guaranteed good resolutions in the time domain, the location space, and the k space. Further parameters are ex =1 ms, in =8 ms, d =6 ms, g = 1.2, Q = 0.05 mV 2 / s, and f = 5 mm, i = 200 mm for = 0.025, f = i = 5 mm for = 1, and f = 200 mm, i = 5 mm for = 40. The abbreviations after the values denote analytical results based on Eq. ͑6͒ ͑analyt.͒ and numerical results based on Eq. ͑3͒ ͑num.͒.

FIG. 4 .

 4 FIG. 4. Response function of the system in Fourier space R͑ , l͒ and the integrand R͑ , l͒C ˜͑l͒ in Eq. ͑6͒ for two values of = ͑a͒, ͑b͒ 1 / 40, and ͑c͒, ͑d͒ 40. Other parameters are taken from Fig. 2.

  Here x krepresents the spike train of the kth neuron, i.e., a sum of ␦ functions. Similar to the previous model, the feedback delay is denoted by d . This feedback model has been modified from ͓17͔ to include the spatial feedback profile F͑j -k͒. Figures 5͑b͒ and 5͑d͒ show the same qualitative increase in power in the gamma range as decreases. We also observe that topographic feedback ͑large 1 / ͒ enhances gamma activity compared to global feedback ͑small 1 / ͒. Moreover,

ACKNOWLEDGMENTS

We thank David Hansel for useful discussions. This research was supported by NSERC Canada.