Dollyane Muret
email: d.muret@ucl.ac.uk

Tamar R Makin
email: t.makin@ucl.ac.uk

The homeostatic homunculus: rethinking deprivation-triggered reorganisation

Homeostatic mechanisms may sustain persistent representation of a missing hand • Recent evidence indicates distributed latent activity across the homunculus • Deprivation uncovers pre-existing latent activity, that can manifest as remapping • Remapping could in some instances correspond to stabilising homeostatic mechanisms Abstract:

While amputation was considered a prominent model for cortical reorganisation, recent evidence highlights persistent representation of the missing hand. We offer a new perspective on the literature of amputation-triggered sensorimotor plasticity, by emphasising the need for homeostasis and emerging evidence of latent activity distributed across the homunculus. We argue that deprivation uncovers pre-existing latent activity, which can manifest as remapping, but that since this activity was already there, remapping could in some instances correspond to functional stability of the system rather than reorganisation. Adaptive behaviour and Hebbian-like plasticity may also play crucial roles in maintaining the functional organisation of the homunculus when deprivation occurs in adulthood or in early development. Collectively, we suggest that the brain's need for stability may underlie several key phenotypes for brain remapping, previously interpreted as consequential to reorganisation. Nevertheless, reorganisation may still be possible, especially when cortical changes contribute to the stability of the system.

BALANCING BETWEEN PLASTICITY AND STABILITY

Seminal work on sensorimotor plasticity has established the notion that the adult cortex is extremely plastic. For example, following amputation of a finger [START_REF] Merzenich | Somatosensory cortical map changes following digit amputation in adult monkeys[END_REF] or deafferentation of an arm [START_REF] Pons | Massive cortical reorganization after sensory deafferentation in adult macaques[END_REF], the deprived primary somatosensory cortex (SI) of monkeys becomes activated by inputs of the cortically neighbouring finger or face, resulting in shifted map boundaries. It has long been assumed that such cortical remapping has direct consequences on perception and action, for example in the context of rehabilitation following stroke or amputation [START_REF] Hara | Brain plasticity and rehabilitation in stroke patients[END_REF][START_REF] Collins | A review of current theories and treatments for phantom limb pain[END_REF]. This notion of a 'soft-wired brain' contrasts with the fact that the brain needs stability and balance to maintain network coherence and cortical dynamics that are essential for its functioning. Indeed, one cannot navigate in a city if its streets keep changing names and directions, as they walk. Similarly, acquisition and preservation of higher-level skills require stable lower sensory representations to draw from. Worse, if not balanced, local plastic changes could theoretically lead to activity overload [START_REF] Hennequin | Inhibitory Plasticity: Balance, Control, and Codependence[END_REF]. Therefore, an appropriate balance between stability and plasticity must be maintained.

Homeostasis corresponds to the ability for a cell or network to detect a perturbation and generate a compensatory response that restores baseline function, thus maintaining stable internal states and balance. At the brain level, homeostatic plasticity often refers to neuronal feedback mechanisms which maintain neural activity within a functional range [START_REF] Turrigiano | The dialectic of hebb and homeostasis[END_REF]. In particular, deprivation has been shown to trigger a rapid modulation of inhibitory networks, followed by a relatively slower mechanism of synaptic scaling in excitatory networks [START_REF] Gainey | Multiple shared mechanisms for homeostatic plasticity in rodent somatosensory and visual cortex[END_REF]. While the first, rapid, process leads to disinhibition that can enable some features of Hebbian plasticity [START_REF] Zenke | The temporal paradox of Hebbian learning and homeostatic plasticity[END_REF], the second, slower, process is mediated by synaptic scaling, which regulates and restores mean firing rate of excitatory neurons to normal levels, associated sometimes with homeostatic adjustment of intrinsic excitability, also in excitatory neurons [START_REF] Gainey | Multiple shared mechanisms for homeostatic plasticity in rodent somatosensory and visual cortex[END_REF]. Here, we define brain homeostasis as the mechanisms that would either strengthen or weaken activity that is respectively above or below what is necessary to maintain baseline function and activity balance at either a local or network level. While homeostatic control of neuronal activity at the cellular level has been widely investigated in the last two decades [START_REF] Vitureira | Homeostatic synaptic plasticity: from single synapses to neural circuits[END_REF][START_REF] Lee | Mechanisms of Homeostatic Synaptic Plasticity in vivo[END_REF], the study of biological mechanisms underlying homeostatic control of neural network function is receiving attention only more recently [START_REF] Wu | Network and cellular mechanisms underlying heterogeneous excitatory/inhibitory balanced states[END_REF][START_REF] Ma | Cortical Circuit Dynamics Are Homeostatically Tuned to Criticality In Vivo[END_REF][START_REF] Barnes | Deprivation-Induced Homeostatic Spine Scaling In Vivo Is Localized to Dendritic Branches that Have Undergone Recent Spine Loss[END_REF]. In this context, development, deprivation and activity-dependent plasticity constitute real challenges to maintaining balance within the sensorimotor system, without over-constraining plasticity that is vital for adaptive changes.

Here we will use the framework of homeostasis to offer a new perspective over classical and recent findings associated with deprivation-triggered plasticity within the human SI. While our focus will be on SI, we will also review relevant studies in the primary motor cortex (M1), in view of their tight functional and anatomical link [START_REF] Kumar | Somatosensory cortex participates in the consolidation of human motor memory[END_REF]. We will first report converging evidence for persistent representation of the missing hand in the deprived cortex of amputees, that we propose may be sustained by homeostatic mechanisms. We will next highlight recent evidence indicating that latent activity across the homunculus is more distributed than previously considered. We will argue that deprivation uncovers this preexisting latent activity, which can manifest as remapping (see Glossary), but as this latent activity was already there, remapping could in some instances correspond to functional stability of the system rather than reorganisation (see Glossary). Finally, we will emphasise the potential importance of activity-dependent plasticity in maintaining the functional organisation of the sensorimotor cortex when deprivation occurs in adulthood (i.e., in amputees), and in early development (i.e., in congenital hand absence). We will suggest that in the latter case, large-scale alteration of the innate sensorimotor topography is possible, but that this form of 'remapping' might be driven by homeostatic preservation of function rather than topography. Collectively, we suggest that the brain's need for stability might be underlying several key phenotypes for brain remapping, previously interpreted as a consequence of reorganisation.

Glossary Homeostasis:

mechanisms that can either strengthen or weaken activity that is respectively above or below what is necessary to maintain baseline function and activity balance at either a local or a network level.

Remapping:

changes to cortical map boundaries or topography, quantified based on activity measures. e.g., activity for one body-part identified in the cortical area of another body-part. Reorganisation: activity changes that introduce new local representational content and readout to downstream brain areas; likely to impact behaviour related to the original or revised function.

Unmasking:

uncovering of latent activity by removal of the dominant activity elicited by the primary input. While generally regarded as an active mechanism, the same outcome may be obtained by passive removal of the primary input without additional updating of activity profiles, if latent activity is indeed more distributed than previously considered.

PERSISTENT REPRESENTATION DESPITE CORTICAL REMAPPING

Contrary to the prevalent view of cortical remapping but consistent with the concept of homeostasis, recent accumulating evidence demonstrates that the canonical representational structure and function of the deprived cortex persist after amputation. This research draws on the lingering sensations arising from the missing limb that affect about 90% of adult upper-limb amputees [START_REF] Weeks | Phantom limb pain: Theories and therapies[END_REF]. By taking advantage of amputees' ability to voluntarily move their phantom hand, the signals elicited both centrally and at the periphery were found to differ from those produced by movement imagery [START_REF] Scaliti | And Yet It Moves: What We Currently Know about Phantom Arm Movements[END_REF]. This experimental approach, combined with advanced fMRI methods, uncovered SI finger maps of the missing hand, even decades after amputation [START_REF] Kikkert | Revealing the neural fingerprints of a missing hand[END_REF]. Canonical hand representational structure was found in all tested amputees, regardless of the vividness of their phantom sensations or pain [START_REF] Wesselink | Obtaining and maintaining cortical hand representation as evidenced from acquired and congenital handlessness[END_REF](Figure 1A). Phantom hand activity was successfully used to decode a range of hand gestures performed with the phantom hand, suggesting a rich representational content of the missing hand in the deprived cortex [START_REF] Bruurmijn | Preservation of hand movement representation in the sensorimotor areas of amputees[END_REF]. Further support for preserved phantom hand representations is provided by assessing the sensations elicited by direct stimulation to the cortex [START_REF] Gunduz | Motor Cortex Reorganization in Limb Amputation: A Systematic Review of TMS Motor Mapping Studies[END_REF] or to the peripheral nerve [START_REF] Strauss | Characterization of multi-channel intraneural stimulation in transradial amputees[END_REF](Figure 1B). Perhaps most strikingly, intracortical micro-stimulation within the SI hand area of a tetraplegic patient elicited tactile sensations localised to the insensate hand, and not any other body-part [START_REF] Flesher | Intracortical microstimulation of human somatosensory cortex[END_REF][START_REF] Flesher | Restored tactile sensation improves neuroprosthetic arm control[END_REF](Figure 1C). Together with the high incidence of phantom sensations, these studies illustrate a high level of functional preservation across the sensorimotor pathway, with stable processing both at the representational and phenomenological levels, consistent with a homeostasis framework.

CORTICAL REMAPPING DESPITE PERSISTENT REPRESENTATION

How can we resolve the classical notion of remapping with this newly accumulating evidence for persistent representation? While these two processes are intuitively contradictory, it is important to point that they are not mutually exclusive, and can in fact co-exist [START_REF] Kikkert | Revealing the neural fingerprints of a missing hand[END_REF][START_REF] Wesselink | Obtaining and maintaining cortical hand representation as evidenced from acquired and congenital handlessness[END_REF]24].

Here we suggest that in the context of the brain's need for homeostasis, these processes are more complementary, in particular when considering the functional organisation of the sensorimotor cortex.

Following on Graziano's pioneering work [START_REF] Graziano | The Organization of Behavioral Repertoire in Motor Cortex[END_REF], recent evidence in non-human primates and human M1 [START_REF] Zeharia | Negative blood oxygenation level dependent homunculus and somatotopic information in primary motor cortex and supplementary motor area[END_REF][START_REF] Willett | Hand Knob Area of Premotor Cortex Represents the Whole Body in a Compositional Way[END_REF][START_REF] Mayer | The Multiple Representations of Complex Digit Movements in Primary Motor Cortex Form the Building Blocks for Complex Grip Types in Capuchin Monkeys[END_REF][START_REF] Huber | Sub-millimeter fMRI reveals multiple topographical digit representations that form action maps in human motor cortex[END_REF] and SI [START_REF] Tal | Positive and Negative Somatotopic BOLD Responses in Contralateral Versus Ipsilateral Penfield Homunculus[END_REF][START_REF] Baldwin | Representations of fine digit movements in posterior and anterior parietal cortex revealed using long-train intracortical microstimulation in macaque monkeys[END_REF] suggests that body-part information is more distributed along the homunculus than commonly thought. Consistent with Penfield's emphasis of high overlap between body parts, this implies that some local activity might be latent relative to a more dominant activity elicited by its primary input (e.g., mouth and feet information in the hand cortical area, see [START_REF] Willett | Hand Knob Area of Premotor Cortex Represents the Whole Body in a Compositional Way[END_REF]) (Figure 2A). If confirmed in SI, this evidence introduces a shift in our conceptualisation from a 'homunculus' with rigid body-part boundaries, to complex maps containing information from different body parts that are co-recruited in our daily life. This new perspective has two main consequences.

Methodologically, considering the presence of latent activity across the homunculus, the way we perform our mapping (e.g., which body parts we decide to map) is going to affect the structure of our maps (Figure 2B). For instance, both electrophysiological (e.g., [START_REF] Merzenich | Somatosensory cortical map changes following digit amputation in adult monkeys[END_REF][START_REF] Pons | Massive cortical reorganization after sensory deafferentation in adult macaques[END_REF]) and neuroimaging studies (e.g., [START_REF] Flor | Phantom-limb pain as a perceptual correlate of cortical reorganization following arm amputation[END_REF][START_REF] Raffin | Primary motor cortex changes after amputation correlate with phantom limb pain and the ability to move the phantom limb[END_REF]) usually engage a limited set of body parts to define maps and remapping in a winner-takes-all manner (i.e., by assigning a neuron/cortical unit to the most responsive body-part among the ones tested). By doing so, other potentially more dominant inputs (e.g. the phantom hand) are ignored, thus creating a 'bias' towards tested body parts (Figure 2B). For example, an apparent remapping obtained by omitting a finger from a finger map analysis strikingly resembles the remapping observed following local anaesthesia of that finger (Wesselink et al, unpublished; Figure 2C). This selection bias is further exacerbated by legacy research, as body parts previously implicated in remapping are more likely to be selected for further testing (e.g. lower face versus upper face in amputees). Therefore, some descriptions of remapping may be misleading.

Conceptually, under the framework of distributed activity, the notion of remapping (Glossary) does not entail reorganisation of inputs. Remapping can result from the elimination of the primary (and most dominant) input via the phenomenon of 'unmasking' (Glossary) as initially suggested by Garraghty and colleagues [START_REF] Garraghty | Pattern of peripheral deafferentation predicts reorganizational limits in adult primate somatosensory cortex[END_REF]. Unmasking is commonly thought to be associated with plasticity, where the release of lateral inhibition allows neighbouring representations to 'invade' the deprived cortex. But rodent studies in SI indicate that this form of disinhibition-driven remapping occurs only transiently, with responses returning towards baseline levels either fully [START_REF] Li | Cortical functional reorganization in response to intact forelimb stimulation from acute to chronic stage in rodent amputation model[END_REF] or partially [START_REF] Albieri | Rapid bidirectional reorganization of cortical microcircuits[END_REF] within a week or two. In the case of long-term remapping, if the underlying activity was already present prior to deprivation, though latent, it stands to reason that homeostatic mechanisms will boost this activity in order to maintain network stability. As such, the fact that deprivation uncovered this activity to the experimenters does not mean that it has impacted the way the brain reads out this activity from the deprived area. In fact, deprivation-triggered remapping, as identified in amputees, has been argued to lack any behavioural relevance [START_REF] Makin | Stability of Sensory Topographies in Adult Cortex[END_REF], further indicating that the local functional processing is relatively unchanged. This notion of stabilised local activity despite input loss is therefore not inconsistent with homeostatic principles.

Figure 2: Evidence for distributed latent activity across the homunculus and its impact on mapping and remapping. A) From [START_REF] Willett | Hand Knob Area of Premotor Cortex Represents the Whole Body in a Compositional Way[END_REF]. Activity recorded in the human M1 hand knob area when a variety of Face, Arm/Hand, Head and Leg/Foot movements were performed. This finding indicates that functional information for specific body parts is distributed across the homunculus B) Model adapted from [START_REF] Smirnakis | Lack of long-term cortical reorganization after macaque retinal lesions[END_REF] illustrating the distribution of latent activity (dashed lines crossing map boundaries) and the way this network is affected after hand amputation (bottom), when excluding (left) and including (right) stabilising inputs from the phantom hand. The unmasking of inputs does not necessarily impact the way the brain reads out this activity from the deprived area. C) From Wesselink et al, unpublished. SI winner-takes-all finger maps, using 7T fMRI, before and after local anaesthesia of digit 2. The map in the middle illustrates a winner-takes-all baseline map calculated when digit 2 is excluded from the baseline analysis. This middle map looks comparable to the post-anaesthesia map (on the right), which suggests that remapping can correspond to experimental uncovering of underlying activity.

CONSIDERING ACTIVITY-DEPENDENT PLASTICITY AS A STABILISING PRINCIPLE

Plasticity in the adult sensorimotor cortex following deprivation has also been suggested to be triggered by contextual changes to the spared input, due to adaptive behaviour. For instance, unilateral arm amputees increasingly rely on their intact hand for daily functioning, and this over-use could relate to increased activity for the intact hand in amputees' (ipsilateral) deprived hand area, resulting in remapping of the intact hand's activity profile [START_REF] Valyear | Interhemispheric transfer of post-amputation cortical plasticity within the human somatosensory cortex[END_REF][START_REF] Makin | Deprivation-related and use-dependent plasticity go hand in hand[END_REF][START_REF] Philip | Compensatory changes accompanying chronic forced use of the nondominant hand by unilateral amputees[END_REF]. In the context of homeostasis, it has been suggested that this intact hand representation may help sustain the persistent representational structure of the missing hand underlying the same cortical territory [START_REF] Kikkert | Revealing the neural fingerprints of a missing hand[END_REF], by providing highly organised inputs to maintain network stability. It is therefore possible that, similar to the case of remapping across cortical neighbours, some of the large-scale remapping observed in amputees (e.g., intact hand into the deprived cortex) may also reflect homeostatic mechanisms increasing latent activity levels to restore balance in the network. Importantly, this process does not necessarily change the representational content of the remapped activity [START_REF] Wesselink | Obtaining and maintaining cortical hand representation as evidenced from acquired and congenital handlessness[END_REF] and thus does not necessarily have any behavioural relevance. It is also possible that behaviour does inform remapping in primary sensory cortex, but only transiently. In the longer-term, higher-order areas may adapt to support behaviour, allowing lower-level representations to return to their initial state. In line with this view, a recent study in twins suggests that while rapid short-term changes are more likely within primary sensory cortical areas, long-term structural changes seem to increase with cortical hierarchy [START_REF] Haak | Plasticity versus stability across the human cortical visual connectome[END_REF].

ACTIVITY-DEPENDENT PLASTICITY IN EARLY DEVELOPMENT BRINGS A NEW PERSPECTIVE TO HOMUNCULUS ORGANISATION AND REMAPPING

While large-scale SI reorganisation might be more limited in adults than originally considered, early life development offers a more favourable environment for activity-dependent plasticity. This is because brain organisation is particularly sensitive to experience and activity during a critical period within development [START_REF] Levelt | Critical-Period Plasticity in the Visual Cortex[END_REF], and because white matter pathways are still under-determined (e.g. pruning). Additionally, homeostatic principles may be different for a developing brain and allow for more reorganisation [START_REF] Nahmani | Adult cortical plasticity following injury: Recapitulation of critical period mechanisms?[END_REF][START_REF] Ribic | Stability in the Face of Change: Lifelong Experience-Dependent Plasticity in the Sensory Cortex[END_REF][START_REF] Frischknecht | Neural ECM molecules in axonal and synaptic homeostatic plasticity[END_REF]. Since hand representation is already established from birth [START_REF] Arcaro | Body map proto-organization in newborn macaques[END_REF], individuals with congenital hand malformations (either unilateral or bilateral -hereafter, one-handers and no-handers, respectively) provide a unique opportunity to examine the potential of SI for remapping and possibly reorganisation.

Studies in these individuals revealed that their sensorimotor missing hand area is lacking an organised hand representation [START_REF] Wesselink | Obtaining and maintaining cortical hand representation as evidenced from acquired and congenital handlessness[END_REF] and is instead activated more than controls by inputs from a range of body parts, such as the feet, mouth and residual arm [START_REF] Hahamy | Remapping in cerebral and cerebellar cortices is not restricted by somatotopy[END_REF][START_REF] Striem-Amit | Plasticity based on compensatory effector use in the association but not primary sensorimotor cortex of people born without hands[END_REF](Figure 3A,B). Importantly, single-pulse TMS to the missing hand M1 area of no-handers elicits motor evoked potentials on the feet, or interferes with foot movements [START_REF] Nakagawa | Cortical reorganization of lower-limb motor representations in an elite archery athlete with congenital amputation of both arms[END_REF][START_REF] Stoeckel | Congenitally altered motor experience alters somatotopic organization of human primary motor cortex[END_REF], demonstrating that the missing hand cortex supports motor control of other body parts that are not necessarily cortical neighbours. This rare demonstration of a causal relationship between local processing in the missing hand cortex and behaviour indicates that reorganisation of motor output could potentially occur, and that this process is not restricted to cortical neighbours. Changes to the SI somatotopy were also found more locally within the foot area, with no-handers showing a detailed map of their toes, structured in a hand-like representational pattern, not found in typically-developed two-handed controls [START_REF] Dempsey-Jones | Organized Toe Maps in Extreme Foot Users[END_REF].

If not exclusively driven or limited by somatotopy, one potential driver for remapping that has been proposed is adaptive behaviour, which is the utilisation of other body parts to substitute for the original function of the missing limb. For example, daily behavioural strategies of one-handers who use their residual arm, mouth and feet to substitute for their missing hand's function were reflected in brain activity and connectivity patterns [START_REF] Makin | Deprivation-related and use-dependent plasticity go hand in hand[END_REF][START_REF] Hahamy | Representation of Multiple Body Parts in the Missing-Hand Territory of Congenital One-Handers[END_REF][START_REF] Hahamy | Normalisation of brain connectivity through compensatory behaviour, despite congenital hand absence[END_REF](Figure 3C), suggesting Hebbian-like plasticity may be at play. Based on this evidence, it has been proposed that remapping may be facilitated or even limited by the functional role of the deprived brain region [START_REF] Hahamy | Representation of Multiple Body Parts in the Missing-Hand Territory of Congenital One-Handers[END_REF], such that only inputs by body parts that engage in the same functional role of the missing limb would be consolidated. This theory of function-dependent remapping (also termed domain-specificity; [START_REF] Mattioni | Categorical representation from sound and sight in the ventral occipito-temporal cortex of sighted and blind[END_REF]) has been contested [START_REF] Striem-Amit | Plasticity based on compensatory effector use in the association but not primary sensorimotor cortex of people born without hands[END_REF][START_REF] Yu | Somatotopic reorganization of hand representation in bilateral arm amputees with or without special foot movement skill[END_REF](Figure 3B) and awaits more direct empirical evidence. Nevertheless, it links the developmental findings for sensorimotor remapping to the broader context of homeostasis, whereby the functional balance in the network is maintained. Indeed, if we consider the sensorimotor homunculus more in terms of clusters of function than in terms of somatotopy, then the territory of a certain body-part region (e.g. the hand area) is in fact responsible for facilitating the function of that body-part (e.g. grasping, manipulating objects) rather than representing the body-part itself. While normally predominantly achieved with a given effector (e.g., the hands), some functions could be supported by other means (e.g., feet, mouth), suggesting that this area would already have the capacity to support these effectors (though in a reduced capacity, compared to the primary effector). Under congenital hand loss, this normally latent activity from effectors with homologous function would be unmasked and further boosted through homeostatic mechanisms. As such, the apparently dramatic remapping described in one/no-handers might be viewed as a network-level adaptation to ensure that the brain develops as stably as possible [START_REF] Defranco | Phantom Sensations : What's a Brain to Do ? A Critical Review of the Re-mapping Hypothesis[END_REF]. Thus, even though the developmental aspects render individuals with congenital hand malformation fundamentally different from amputees, these two populations may comply to similar homeostatic principles that stabilise latent activity, which may be more distributed than initially thought. From [START_REF] Hahamy | Remapping in cerebral and cerebellar cortices is not restricted by somatotopy[END_REF]. Univariate contrast maps between one-handers and controls during residual/nondominant arm (one-handers/controls), lips, feet, and intact/dominant hand movements, projected onto an inflated surface of the hemisphere contralateral to the missing hand. In each of the arm, lips, and feet (but not intact hand) conditions, one-handers showed increased activity in the missing hand cortex compared with controls. Yellow/blue contours represent the hand/lip ROIs, respectively. B) From [START_REF] Striem-Amit | Plasticity based on compensatory effector use in the association but not primary sensorimotor cortex of people born without hands[END_REF]. Sensorimotor maps obtained in control participants, illustrating the hand region (white contour) used to extract the underlying cortical activity in congenital no-handers. As shown in the bottom, increased activity was observed for all body parts compared to controls. C) From [START_REF] Hahamy | Representation of Multiple Body Parts in the Missing-Hand Territory of Congenital One-Handers[END_REF]. Illustration of the adaptive behaviour displayed by congenital one-handers when asked to open a bottle. The body parts used to compensate for the absence of the hand (i.e., residual arm, lips and legs) are those exhibiting increased activity in the missing hand cortex shown in A).

CONCLUDING REMARKS

Here we offer a new perspective on the literature of deprivation-triggered sensorimotor plasticity, by emphasising the brain's need for homeostasis and the presence of distributed latent activity across the homunculus. We suggest that these two guiding principles can unify disparate, and often intuitively contradictory evidence, relating to the sensorimotor system's response to deprivation following amputation. Persistent missing-hand representation and activity-dependent plasticity may both help maintain the functional organisation of the sensorimotor cortex following deprivation. Simply put -inputs that contribute to the stability of the sensorimotor system organisation could be consolidated via Hebbian plasticity while other inputs are negated by homeostatic mechanisms. Importantly, under this framework, the notion of distributed latent activity implies that rather than triggering reorganisation via disinhibition, unmasking may trigger homeostatic mechanisms that help preserve the underlying pre-existing functional structure and processing of sensorimotor cortex in case of deprivation. As such, uncovering this latent activity may not necessarily alter the representational content in the deprived cortex or relate to behaviour. Thus, rather than being associated with reorganisation, remapping may (under some circumstances) correspond to stability of the system. Nevertheless, it is important to emphasise that under our suggested framework, reorganisation at a finer scale is not impossible to achieve, especially in the context of learning, so long as such changes do not threaten the stability of the network at a larger scale. Finally, using development as an extreme model for remapping, we showed that large-scale remapping may be possible, albeit under special circumstances.

Here we suggest that although activity profiles might be modulated relative to typical development, the overall functional organisation of the brain is maintained, when considering homuncular organisation as guided by function rather than by specific inputs. However, the potential behavioural relevance of this large-scale remapping still needs to be clarified. This process could therefore also reflect the brain's overarching drive for stability across lifespan. Future efforts for restoring hand function, such as brain-computer interfaces for artificial limb control, biological hand transplantation, stroke rehabilitation and hand augmentation could harness and benefit from this functional and structural stability.

Figure 1 :

 1 Figure 1: Evidence for persistent representation. A) From[START_REF] Wesselink | Obtaining and maintaining cortical hand representation as evidenced from acquired and congenital handlessness[END_REF]. Representational structure of the non-dominant hand in controls (upper panel) and of the missing (phantom) hand in upperlimb amputees (lower panel) as measured by fMRI representational similarity analysis. The structure is portrayed by two-dimensional projection using multi-dimensional scaling. Dissimilarity is reflected by the distance in the two dimensions; individual digits are indicated by different colours (see insert); and ellipses show the between-subject standard error after Procrustes alignment. The representational structure of the phantom hand was statistically similar relative to controls. B) From[START_REF] Strauss | Characterization of multi-channel intraneural stimulation in transradial amputees[END_REF]. Distribution of phantom sensations evoked by intraneural stimulation in an example amputee. The implantation of electrodes on the median (M1-2) and ulnar (U1-2) nerves is shown on the right schema. The colour code used to depict evoked sensations corresponds to specific electrodes (see insert). Peripheral stimulation causes sensations on the missing (phantom) hand. C) Adapted from[START_REF] Flesher | Intracortical microstimulation of human somatosensory cortex[END_REF]. Projected field maps for the hand of a tetraplegic patient in response to intracortical micro-

Figure 3 :

 3 Figure 3: Remapping observed in individuals born with congenital hand malformations. A)From[START_REF] Hahamy | Remapping in cerebral and cerebellar cortices is not restricted by somatotopy[END_REF]. Univariate contrast maps between one-handers and controls during residual/nondominant arm (one-handers/controls), lips, feet, and intact/dominant hand movements, projected onto an inflated surface of the hemisphere contralateral to the missing hand. In each of the arm, lips, and feet (but not intact hand) conditions, one-handers showed increased activity in the missing hand cortex compared with controls. Yellow/blue contours represent the hand/lip ROIs, respectively. B) From[START_REF] Striem-Amit | Plasticity based on compensatory effector use in the association but not primary sensorimotor cortex of people born without hands[END_REF]. Sensorimotor maps obtained in control participants, illustrating the hand region (white contour) used to extract the underlying cortical activity in congenital no-handers. As shown in the bottom, increased activity was observed for all body parts compared to controls. C) From[START_REF] Hahamy | Representation of Multiple Body Parts in the Missing-Hand Territory of Congenital One-Handers[END_REF]. Illustration of the adaptive behaviour displayed by congenital one-handers when asked to open a bottle. The body parts used to compensate for the absence of the hand (i.e., residual arm, lips and legs) are those exhibiting increased activity in the missing hand cortex shown in A).

Acknowledgements: This work was supported by an ERC Starting Grant (715022 EmbodiedTech) and a Wellcome Trust Senior Research Fellowship (215575/Z/19/Z), awarded to TRM. We thank Avital Hahamy, Daan Wesselink, Hannes Saal, Ella Striem-Amit and Esther Kuehn for helpful feedback on the manuscript and Arabella Bouzigues for proof reading.


Annick Loschetter
email: a.loschetter@brgm.fr

Christophe Kervévan

Rowena Stead

Thomas Le

Thomas Le Guénan

Water-driven

Geothermal energy production and CCS (Carbon Capture and Storage) represent promising technological solutions to help mitigate climate change and aid the current global energy crisis. In recent years, the number of concepts that propose to combine and mutualize these technologies has risen dramatically. While a number of concepts (notably CPG and CO 2 -EGS) use supercritical CO 2 as the heat vector, another promising route for hybridization is to inject dissolved CO 2 in the geothermal brine. This is the focus of our current work. An extensive literature review was carried out of the concepts, complemented by interviews of some of the developers.

A few concepts are still theoretical (only described in the literature), but most technological ones are on the way to pilot/demonstration projects at progressively increasing scale. The main projects and their associated sites are: CO 2 -DISSOLVED technology, with potential sites identified in the Paris basin (France); AAT-G / Cleag technology, with a site in Croatia; Related projects CarbFix, GECO and SUCCEED, with sites in Hellisheidi (Iceland), Nesjavellir (Iceland), Bochum (Germany), Kızıldere (Turkey), Castelnuovo in Italy (as case study due to permitting issues); Reinjection of CO 2 from geothermal brines at Ngatamariki and Te Huka sites in New Zealand.

Despite similarities, these solutions are differentiated by their purpose:

either to store CO 2 from an external industrial emitter (notably for the CO 2 -Dissolved concept), thus bringing a contribution to CCS, or to reinject CO 2 emitted by CO 2 -rich brine during geothermal exploitation, thus bringing geothermal to near-zero emissions.

Because of the CO 2 solubility limit in brine, the performance of heat extraction is generally higher than that of CO 2 storage. For instance the CO 2 -DISSOLVED technology is particularly well-suited to small CO 2 industrial emitters (ca. <150,000 t CO 2 /year). Unlike concepts using supercritical CO 2 , those using dissolved CO 2 can be deployed at much lower depths (no need to exceed the supercritical point). The concepts still need a tight caprock, but the high solubility trapping represents a lower risk of leakage. The geothermal system behaves mainly as a water-driven system, but the adjunction of dissolved CO 2 can in some cases increase thermo-hydrological

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Acknowledgements: This work is in part taken from a study published in the report 'IEAGHG, "Prospective integration of Geothermal Energy with Carbon Capture and Storage (CCS)", 2023-02, August 2023'. We are grateful to IEAGHG, especially to Nicola Clarke, for proposing and funding this topic and for interesting scientific discussions and debates as part of this work.

A number of challenges still need to be addressed, including complexity of regulations and validation of some technical aspects. Besides, considering the variety of underground configurations, there is no turnkey solution, which might hamper the economics of such smallscale projects.


Patrick Cassam-Chenaï
email: cassam@unice.fr

Simple Conditions for the Transformation of Dynamical Coordinates into Canonical Ones in Hamiltonian Dynamics

We obtain conditions, which when fulfilled, permit to transform the coordinates of a dynamical system into pairs of canonical ones for some Hamiltonian system. These conditions, restricted to the class of coordinate transformations which act on each coordinate independently, are greatly simplified. However, they are surprisingly successful in defining canonical coordinates and an associated Hamiltonian for several test examples. So, a method is proposed to exploit these simple transformations in a systematic manner.

Introduction

A Hamiltonian dynamical system has specific properties, which can be taken advantage of in its study [START_REF] Mackay | Hamiltonian dynamical systems : A reprint selection compiled and introduced[END_REF]. General, N -dimensional, Poisson brackets permit to obtain generalized Hamiltonians [START_REF] Olver | Applications of Lie Groups to Differential Equations[END_REF]. However, quantization of the latter can be ambiguous [START_REF] Yan | [END_REF]. Here, we will take the term "Hamiltonian" in the restricted sense of a function giving Hamilton equations for pairs of conjugate canonical coordinates. Such canonical Hamiltonian systems can be quantized by replacing Poisson brackets by commutators [START_REF] Dirac | [END_REF].

Darboux's theorem or its generalization [5] insures that locally there always exists a canonical representation of a Hamiltonian in the generalized sense. In this work, we aim at finding directly this canonical representation. It is well-known that it is not possible to change the canonical or non-canonical nature of a dynamical system with a linear transformation of the coordinates. In this work, we show how it is feasible with a nonlinear but simple one, by just writing equalities [START_REF] Avez | Differential calculus[END_REF], that must be satisfied according to Schwartz theorem after transformation. The method is applied succesfully to several classical models, in spite of the quite severe restriction we make, that the transformations act on each coordinate independently.

As straightforward and simple as the method might be, we have not found such proposal in the literature. For example, the Lotka-Volterra model in 2-dimension has been ]ocnmp[ Patrick Cassam-Chenaï extensively studied and the Hamiltonian we obtain with our approach is known [START_REF] Kerner | [END_REF]8]. However, the systematic derivation of an Hamiltonian by defining an appropriate generalized Poisson bracket [9] or by time rescaling [10], give different Hamiltonian functions as there is no change of the dynamical system coordinates. Volterra's own construction preserves the canonical Poisson bracket, but introduce extra-variables, and the Hamiltonian he obtained, though formally close to ours, gives more complicated Hamilton-Jacobi equations [START_REF] Volterra | Leçons sur la Théorie Mathématique de la Lutte pour vie[END_REF][START_REF] Duarte | [END_REF]. In addition, it appears that our transformed coordinates can be of physical interest for the description of the system. So, we propose an algorithm based on Schwartz theorem equalities, to deal with general systems, which searches for, (at least,) a subset of coordinates amenable to be turned into canonically conjugate ones, satisfying a set of Hamilton's equations.

The article is organized as follows: First we present the method in two-dimension and illustrate its application on two case examples. Then, we generalize it to any finite dimension, and apply it to the Kermack-McKendrick model, before concluding.

2 Transformation of a dynamical system in R 2

Theory

We consider a dynamical system in 2 real coordinates in an open set U ⊆ R 2 , defined by a mapping f :

U -→ R 2 , that is to say, f = (f 1 , f 2 ), ∀i ∈ {1, 2} f i : U -→ R is smooth enough (often considered to be C ∞ ,
however this is not necessay for our purpose, it may also depends upon the time variable, but we will omit this too), and, ẋi = f i (x 1 , x 2 ).

We do not know a priori whether the system is Hamiltonian or not. Hoping it is Hamiltonian, we look for a diffeomorphism h : U -→ R 2 , whose components h 1 and h 2 define two new coordinates,

q = h 1 (x 1 , x 2 ) (1) p = h 2 (x 1 , x 2 ). (2)
that satisfy Hamilton's equations for some Hamiltonian function, H,

q = ∂H ∂p , (3) ṗ
= - ∂H ∂q . (4)
We are going to obtain necessary conditions for this problem to admit solutions. Note that, if the system is Hamiltonian, then, there will be infinitely many solutions. We denote the inverse mapping h -1 by g = (g 1 , g 2 ),

x 1 = g 1 (q, p) , (5)
x 2 = g 2 (q, p) .

The local inversion theorem [START_REF] Avez | Differential calculus[END_REF] insures that for (x 1 , x 2 ) = g(q, p) such that (q, p) ∈ h(U ),

we have the relation between the differentials: Dh(x 1 , x 2 ) = [Dg(q, p)] -1 , so that:

∂h 1 (x 1 , x 2 ) ∂x 1 = ∂g 2 ∂p ∂g 1 ∂q ∂g 2 ∂p -∂g 1 ∂p ∂g 2 ∂q , (7)
∂h 1 (x 1 , x 2 ) ∂x 2 = -∂g 1 ∂p ∂g 1 ∂q ∂g 2 ∂p -∂g 1 ∂p ∂g 2 ∂q , ( 8
)
∂h 2 (x 1 , x 2 ) ∂x 1 = -∂g 2 ∂q ∂g 1 ∂q ∂g 2 ∂p -∂g 1 ∂p ∂g 2 ∂q , ( 9
)
∂h 2 (x 1 , x 2 ) ∂x 2 = ∂g 1 ∂q ∂g 1 ∂q ∂g 2 ∂p -∂g 1 ∂p ∂g 2 ∂q . ( 10
)
Injecting expressions (1) and (2) in Eq.( 3) and using the chain rule, we obtain

∂H ∂p = ∂h 1 (x 1 , x 2 ) ∂x 1 ẋ1 + ∂h 1 (x 1 , x 2 ) ∂x 2 ẋ2 = ∂h 1 (x 1 , x 2 ) ∂x 1 f 1 (g 1 (q, p), g 2 (q, p)) + ∂h 1 (x 1 , x 2 ) ∂x 2 f 2 (g 1 (q, p), g 2 (q, p)) , (11)
and similarly, using Eq.( 4),

- ∂H ∂q = ∂h 2 (x 1 , x 2 ) ∂x 1 f 1 (g 1 (q, p), g 2 (q, p)) + ∂h 2 (x 1 , x 2 ) ∂x 2 f 2 (g 1 (q, p), g 2 (q, p)) . (12)
Substituting Eqs.( 7)- (10) into Eqs. ( 11) and ( 12), we obtain the Hamilton equations,

∂H ∂p = f 1 (g 1 (q, p), g 2 (q, p)) • ∂g 2 ∂p -∂g 1 ∂p • f 2 (g 1 (q, p), g 2 (q, p)) ∂g 1 ∂q ∂g 2 ∂p -∂g 1 ∂p ∂g 2 ∂q , (13)
- ∂H ∂q = ∂g 1 ∂q • f 2 (g 1 (q, p), g 2 (q, p)) -f 1 (g 1 (q, p), g 2 (q, p)) • ∂g 2 ∂q ∂g 1 ∂q ∂g 2 ∂p -∂g 1 ∂p ∂g 2 ∂q . ( 14
)
We can either (i) integrate one of these equation and insert the expression obtained for H into the other one, (assuming that g is of class C 2 , so that h is also C 2 ) [START_REF] Avez | Differential calculus[END_REF], to derive a relation between f 1 , f 2 , and g (or more conveniently h), or, (ii) (assuming that, in addition, the f i 's have first partial derivatives and H second partial derivatives), we can use Schwartz's theorem [START_REF] Avez | Differential calculus[END_REF] for the same purpose. Noting ī = 2 if i = 1 and ī = 1 if i = 2, and using the inverse of Eqs.( 7)-( 10), we obtain, in the original coordinates, the following necessary condition for the existence of a coordinate transformation, h, and a Hamiltonian function, H, such that the Hamilton's equations ( 3) and ( 4) are satisfied:

i∈{1,2} ∂f i ∂x i + f i • ∂ 2 h i ∂x 2 i ∂hī ∂xī -∂ 2 h i ∂x 1 ∂x 2 ∂hī ∂x i - ∂ 2 hī ∂x 2 i ∂h i ∂xī + ∂ 2 hī ∂x 1 ∂x 2 ∂h i ∂x i ∂h 1 ∂x 1 ∂h 2 ∂x 2 -∂h 1 ∂x 2 ∂h 2 ∂x 1 = 0 . ( 15
)
The condition is only necessary, because Schwartz's theorem only asserts that, if a differential form is exact, then it is closed i.e. satisfies Eq. [START_REF] Plank | [END_REF]. The condition will be sufficient

]ocnmp[

Patrick Cassam-Chenaï
if the open set U is simply connected according to Poincaré lemma. Then, if it is not fulfilled, one can deduce from Darboux's theorem that the system is not Hamiltonian. We remark at this stage that, if the h i 's are linear, their second partial derivatives cancel out and we are left with the first term within the sum over i. That is to say, we retrieve the condition for the differential form to be closed in its original coordinates:

∂f 1 (x 1 , x 2 ) ∂x 1 + ∂f 2 (x 1 , x 2 ) ∂x 2 = 0 . (16)
Conversely, when this is not zero but condition ( 15) is fulfilled, then, necessarily, the second terms within the sum over i are not all zero. In particular, if we manage to find some h 1 , h 2 such that, for each i ∈ {1, 2}, the second term cancels the first one within the sum, the differential form will be closed.

To go further with this, we may consider the expansion of the h i (x 1 , x 2 )'s on a set of product functions:

h i (x 1 , x 2 ) = j 1 ,j 2 λ j 1 ,j 2 i χ j 1 (x 1 )χ j 2 (x 2 )
, each factor belonging, for example, to a fixed Hilbertian basis set (χ j (x)) j . We can limit the set of coefficients λ j 1 ,j 2 i to deal with, by truncating this expansion. We may look for one-term expansions of the form

h i (x 1 , x 2 ) = ψ 1 i (x 1 )ψ 2 i (x 2
) and solve Eq.( 15) for the four functions

ψ 1 1 , ψ 2 1 , ψ 1 2 , ψ 2 2 .
Hereafter, we make the even more drastic assumption that each coordinate is transformed independently of the others, that is to say, h i only depends upon x i , so that ∂h i xī = 0. This simplifies greatly the general equations, ( 13) and ( 14), which now become,

∂H ∂p = f 1 (g 1 (q), g 2 (p)) g ′ 1 (q) , (17)
- ∂H ∂q = f 2 (g 1 (q), g 2 (p)) g ′ 2 (p) , (18)
as well as Eq.( 15),

∂f 1 (x 1 , x 2 ) ∂x 1 + h ′′ 1 (x 1 ) h ′ 1 (x 1 ) • f 1 (x 1 , x 2 ) + ∂f 2 (x 1 , x 2 ) ∂x 2 + h ′′ 2 (x 2 ) h ′ 2 (x 2 ) • f 2 (x 1 , x 2 ) = 0 . (19)
In the next sections, we will apply the necessary condition (19) to two different cases and show its practicality to determine the freely adjustable functions h 1 and h 2 .

Volterra's model as a test example

The first example appearing in Ref. [START_REF] Chazottes | Differential Equations -An invitation through embedded visual interactive digital experiments[END_REF], of a dynamical system that is non-canonical in its original coordinates, is Volterra's model of sharks and sardines [START_REF] Volterra | Leçons sur la Théorie Mathématique de la Lutte pour vie[END_REF]:

ẋ1 = a x 1 -b x 1 x 2 , ( 20
) ẋ2 = -c x 2 + d x 1 x 2 , (21)
with a, b, c, d ∈ R + * . In these variables, assuming that ẋ1 = ± ∂H ∂x 2 and ẋ2 = ∓ ∂H ∂x 1 , Schwartz's theorem applied to H, would not be satisfied since (abx 2 ) = ±(cdx 1 ).

Eq.( 19) gives the condition:

(a -b x 2 ) 1 + h ′′ 1 (x 1 ) h ′ 1 (x 1 ) • x 1 + (-c + d x 1 ) 1 + h ′′ 2 (x 2 ) h ′ 2 (x 2 ) • x 2 = 0 . ( 22
)
The condition will be clearly fulfilled if,

h ′′ 1 (x 1 ) h ′ 1 (x 1 ) = - 1 x 1 , h ′′ 2 (x 2 ) h ′ 2 (x 2 ) = - 1 x 2 .
These are easy to integrate, recognizing logarithmic derivatives on both sides, ln(h ′ 1 (x 1 )) = ln(

1 x 1 ) + c 1 , ln(h ′ 2 (x 2 )) = ln( 1 x 2 ) + c 2 .
Choosing c 1 = c 2 = 0, and integrating a second time with zero as integration constants, we finally define the new variables in the following way:

q = h 1 (x 1 ) = ln(x 1 ) , (23)
p = h 2 (x 2 ) = ln(x 2 ) , (24)
that is to say,

x 1 = exp(q) , (25)
x 2 = exp(p) . (26)
The canonical Hamiltonian is obtained from Hamilton's equations ( 17) and ( 18),

H V olterra = a • p -b • exp(p) + c • q -d • exp(q) . (27)
Not surprisingly for such a popular model, Hamiltonian's forms are already known [9,[START_REF] Duarte | [END_REF][START_REF] Glansdorff | Thermodynamic theory of structure, stability and fluctuations[END_REF][START_REF] Kerner | [END_REF]8,[START_REF] Plank | [END_REF]16,17,[START_REF] Baigent | Dynamical and Complex Systems[END_REF]. However, the derivation of Eq.( 27) is particularly straightforward with our approach. The number of coordinates is still two, unlike in Volterra's work [START_REF] Duarte | [END_REF][START_REF] Baigent | Dynamical and Complex Systems[END_REF]. These coordinates are conjugate for the canonical Poisson bracket, unlike in [9,[START_REF] Plank | [END_REF]10] where a generalized Poisson bracket is used or appear from a time coordinate rescaling. The new coordinates are physically relevant: the original ones would evolve exponentially, if they were not coupled, so, it makes sense to describe the system on a logarithmic scale. This is what is achieved thanks to the new coordinates.

A classical system derived from a forced, two-level, quantum model

Consider the following dynamical system derived from a forced, two-level, quantum model [19]:

ẋ1 = -sin(x 2 ) , ( 28
) ẋ2 = B -cot(x 1 ) cos(x 2 ), (29)
where x 1 and x 2 are two real dynamical coordinates, and B a constant. This system is not canonical, for, suppose there exists H such that ẋ1 = ± ∂H ∂x 2 and ẋ2 = ∓ ∂H ∂x 1 , the right hand sides of Eqs. ( 28) and ( 29) being differentiable, we can search for an H that is derivable twice. Then, Schwartz's theorem permits to conclude at the non-existence of such H, since

∂ sin(x 2 ) ∂x 1
= 0 and ∂B-cot(x 1 ) cos(x 2 )

∂x 2 = cot(x 1 ) sin(x 2 ) = 0.
]ocnmp[ Patrick Cassam-Chenaï Equation ( 15) applied to this system gives, cot(x 1 ) -

h ′′ 1 (x 1 ) h ′ 1 (x 1 ) sin(x 2 ) + (B -cot(x 1 ) cos(x 2 )) h ′′ 2 (x 2 ) h ′ 2 (x 2 ) = 0 . ( 30
)
This can be satisfied by taking h 2 linear, and

h ′′ 1 (x 1 ) h ′ 1 (x 1 ) = cot(x 1
). We recognize again logarithmic derivatives and integrate, this gives ln(h ′ 1 (x 1 )) = ln(sin(x 1 )) + c. The simplest choice of integration constants leads to h 2 (x 2 ) = x 2 and h 1 (x 1 ) = cos(x 1 ). Therefore, we define the new coordinates as

q = cos(x 1 ) (31) p = x 2 , , (32)
and the dynamical system becomes:

q = 1 -q 2 sin(p) (33) ṗ = B - q 1 -q 2 cos(p). (34)
We easily verify that Eqs. ( 33) and ( 34) are the Hamilton-jacobi equations for the Hamiltonian,

H = -Bq -1 -q 2 cos(p) . (35)
3 Generalization to a dynamical system in R N

Theory

We now consider a dynamical system in N real coordinates:

∀i ∈ {1, • • • , N } ẋi = f i (x 1 , • • • , x N ) . ( 36
)
If N is odd, one can always add an extra dummy coordinate, which is constant and whose initial value is zero. So, without loss of generality, we can assume N even: N = 2n, and search for a canonical Hamiltonian H, and a transformation of the initial variables, x 1 , . . . , x 2n into two sets {q 1 ,

• • • , q n }, {p 1 , • • • , p n }, such that: ∀i ∈ {1, • • • , n}, qi = ∂H ∂p i (37) ṗi = - ∂H ∂q i . (38)
The general equations ( 13), ( 14) and ( 15) can easily be generalized to a diffeomorphism h(x 1 , • • • , x 2n ), if the 2n coordinates are partitionned into pairs {x i 1 , x i 2 }, and if h acts on each pair independently of the others, that is to say, if for each pair (i 1 , i 2 ), it has a corresponding pair of components h j 1 and h j 2 , which only depend upon the two coordinates x i 1 and x i 2 . Then, without loss of generality, the coordinates and the components can be reordered, so that

∀i ∈ {1, • • • , n}, q i = h 2i-1 (x 2i-1 , x 2i ) ( 39
)
p i = h 2i (x 2i-1 , x 2i ) . (40)
A fortiori, if each h i only depends upon x i , Eqs. ( 17), ( 18) and ( 19) can be generalized. This will give as many necessary conditions of the form (19) to be satisfied simultaneously, as the number of canonically conjugate disjoint pairs to be found. An example is given in the next section.

Kermack-McKendrick's model

The Kermack-McKendrick's model is a dynamical system for 3 real coordinates of the form,

ẋ1 = -rx 1 x 2 (41) ẋ2 = +rx 1 x 2 -ax 2 (42) ẋ3 = +ax 2 , (43)
with a, r ∈ R + * , to which we add a dummy one:

ẋ4 = 0 with x 4 (0) = 0 . ( 44
)
We easily calculate:

∂f 1 (x 1 , • • • , x 4 ) ∂x 1 + d ln(h ′ 1 (x 1 )) dx 1 • f 1 (x 1 , • • • , x 4 ) = -rx 2 1 + d ln(h ′ 1 (x 1 ) dx 1 • x 1 ∂f 2 (x 1 , • • • , x 4 ) ∂x 2 + d ln(h ′ 2 (x 2 )) dx 2 • f 2 (x 1 , • • • , x 4 ) = (rx 1 -a) 1 + d ln(h ′ 2 (x 2 ) dx 2 • x 2 ∂f 3 (x 1 , • • • , x 4 ) ∂x 3 + d ln(h ′ 3 (x 3 )) dx 3 • f 3 (x 1 , • • • , x 4 ) = ax 2 d ln(h ′ 3 (x 3 ) dx 3 ∂f 4 (x 1 , • • • , x 4 ) ∂x 4 + d ln(h ′ 4 (x 4 )) dx 4 • f 4 (x 1 , • • • , x 4 ) = 0.
These four quantities cancel out independently, if one takes: h 1 (x 1 ) = ln(x 1 ), h 2 (x 2 ) = ln(x 2 ), h 3 (x 3 ) = x 3 , h 4 (x 4 ) = x 4 , then, any two pairs can be canonically conjugate for a Hamiltonian system. Choosing q 1 = h 1 (x 1 ), p 1 = h 2 (x 2 ), q 2 = h 3 (x 3 ), p 2 = h 4 (x 4 ) and integrating the Hamilton equations we obtain the Hamiltonian, H Kermack-M cKendrick = -r exp q 1 + aq 1 + (ap 2r) exp p 1 , which gives the correct equations, given that, solving the Hamilton's equations, p 2 = x 4 = x 4 (0) = 0.

Concluding remarks

Since, it is possible that not all but only a subset of coordinates fulfills the necessary conditions imposed by the Schwartz equalities, we propose the following method to detect one or more pairs of variables amenable to be transformed into conjugate canonical variables following a canonical Hamiltonian dynamics.

(i) Calculate ∀i ∈ {1, • • • , N } the f i -dependent part of ∂f i (x 1 ,••• ,x N ) ∂x i + d ln(h ′ i (x i )) dx i •f i (x 1 , • • • , x N ) (ii)
For all i < N , and j > i, try to find h i and h j such that Eq. ( 19) is satisfied for the pair (i, j). In case of success, add pair (i, j) to a set, C, of potentially conjugate pairs. (iii) Choose a maximal subset of disjoint pairs of C, and transform the associated variables using the corresponding h functions.

]ocnmp[ Patrick Cassam-Chenaï Following such a path, it is possible to transform the three non-canonical systems considered in this work, namely a classical system derived from a forced two-level quantum model, Volterra's model and Kermack-McKendrick's model into canonical ones, in a very simple and straightforward fashion.

There is a lot of freedom in the choice of the two one-variable functions, h 1 and h 2 , to fulfill constraint (19). So, we can expect that a large class of dynamical systems should be convertible into canonical ones, even within the severe restriction that the variables be transformed independently of each others.

Of course, not all dynamical systems are supposed to be amenable to our treatment. In particular, one could think of non-conservative systems, such as the damped pendulum, ẋ1 = x 2 , (45) ẋ2 = -ω 2 sin(x 1 )µ x 2 .

(46

)
or the harmonic oscillator with friction (which is just its linearized version obtained by substituting x 1 to sin(x 1 )). However, in these particular cases, and in fact, for any system with a force deriving from a potential f (x 1 ) = -dV (x 1 )

dx 1
with a damping term of the form -µ(t) ẋ1 , ẋ1 = x 2 , (47

) ẋ2 = - dV (x 1 ) dx 1 -µ(t) x 2 , (48)
one can easily extend our method by using time-dependent transformations. More explicitly, one retrieves the original, non-autonomous, non conservative, dynamical system by applying canonical Eqs. ( 3) and ( 4) to the conjugate coordinates q, p and the Hamiltonian H defined by the following equations:

q = h 1 (x 1 ) = x 1 , (49)
p = h 2 (x 2 ) = exp t 0 µ(t ′ )dt ′ x 2 , (50)
H = exp - t 0 µ(t ′ )dt ′ 2 p 2 + exp t 0 µ(t ′ )dt ′ V (q). (51)
So, in fact, the applicability of our proposed method goes beyond what common belief would assume.

Acknowledgements

The Hamiltonian of the forced, two-level, quantum system was brought to our attention by Prof. Gilbert Reinisch. We acknowledge fruitful discussions with Profs. Zhiyan Zhao and Didier Clamond from Université Côte d'Azur. We are grateful to the referee for suggesting many improvements to our manuscript.