Section: New Results

Macroscopic limits of stochastic neural networks and neural fields

Pulsatile localized dynamics in delayed neural-field equations in arbitrary dimension

Participants : Jonathan Touboul, Grégory Faye [EHESS] .

Neural field equations are integro-differential systems describing the macroscopic activity of spatially extended pieces of cortex. In such cortical assemblies, the propagation of information and the transmission machinery induce communication delays, due to the transport of information (propagation delays) and to the synaptic machinery (constant delays). We have investigated the role of these delays on the formation of structured spatiotemporal patterns for these systems in arbitrary dimensions [19] . We have focused on localized activity, either induced by the presence of a localized stimulus (pulses) or by transitions between two levels of activity (fronts). Linear stability analysis allows to reveal the existence of Hopf bifurcation curves induced by the delays, along different modes that may be symmetric or asymmetric. We show that instabilities strongly depend on the dimension, and in particular may exhibit transversal instabilities along invariant directions. These instabilities yield pulsatile localized activity, and depending on the symmetry of the destabilized modes, either produce spatiotemporal breathing or sloshing patterns.

Limits and dynamics of randomly connected neuronal networks

Participants : Cristóbal Quiñinao [CIRB] , Jonathan Touboul.

Networks of the brain are composed of a very large number of neurons connected through a random graph and interacting after random delays that both depend on the anatomical distance between cells. In order to comprehend the role of these random architectures on the dynamics of such networks, we have analyzed the mesoscopic and macroscopic limits of networks with random correlated connectivity weights and delays [35] . We have addressed both averaged and quenched limits, and shown propagation of chaos and convergence to a complex integral McKean-Vlasov equations with distributed delays. We have then instantiated a completely solvable model illustrating the role of such random architectures in the emerging macroscopic activity. We have particularly focused on the role of connectivity levels in the emergence of periodic solutions.

The propagation of chaos in neural fields

Participant : Jonathan Touboul.

We have considered the problem of the limit of bio-inspired spatially extended neuronal networks including an infinite number of neuronal types (space locations), with space-dependent propagation delays modeling neural fields [24] . The propagation of chaos property is proved in this setting under mild assumptions on the neuronal dynamics, valid for most models used in neuroscience, in a mesoscopic limit, the neural-field limit, in which we can resolve the quite fine structure of the neuron activity in space and where averaging effects occur. The mean-field equations obtained are of a new type: they take the form of well-posed infinite-dimensional delayed integro-differential equations with a nonlocal mean-field term and a singular spatio-temporal Brownian motion. We have also shown how these intricate equations can be used in practice to uncover mathematically the precise mesoscopic dynamics of the neural field in a particular model where the mean-field equations exactly reduce to deterministic nonlinear delayed integro-differential equations.

Spatially extended networks with singular multi-scale connectivity patterns

Participant : Jonathan Touboul.

In [24] , we took care of a number of technical difficulties arising in the description of large-scale systems that are spatially extended. The organization of neurons in space (within cortical columns) and their interactions (fully connected networks) were relatively far from what is known of the anatomy of neuronal networks. In [25] , we have further taken into account the fine and macroscopic structure of the cortex, which is a very large network characterized by a complex connectivity including at least two scales. On the microscopic scale, the interconnections are non-specific and very dense, while macroscopic connectivity patterns connecting different regions of the brain at larger scale are extremely sparse. This motivates to analyze the behavior of networks with multiscale coupling, in which a neuron is connected to its $v\left(N\right)$ nearest-neighbors where $v\left(N\right)=o\left(N\right)$, and in which the probability of macroscopic connection between two neurons vanishes. These are called singular multi-scale connectivity patterns. We have introduced a class of such networks and derived their continuum limit. We show convergence in law and propagation of chaos in the thermodynamic limit. The limit equation obtained is an intricate non-local McKean-Vlasov equation with delays which is universal with respect to the type of micro-circuits and macro-circuits involved.

Index Distribution of the Ginibre Ensemble

Participants : Romain Allez [Stastlab, Cambridge University] , Gilles Wainrib [ENS] , Jonathan Touboul.

Complex systems, and in particular random neural networks, are often described by randomly interacting dynamical systems with no specific symmetry. In that context, characterizing the number of relevant directions necessitates fine estimates on the Ginibre ensemble. We have computed analytically the probability distribution of the number of eigenvalues $N_R$ with modulus greater than $R$ (the index) of a large $N\times N$ random matrix in the real or complex Ginibre ensemble [15] . We have shown that the fraction $N_R/N=p$ has a distribution scaling as $exp(-\beta N^2{\psi }_R\left(p\right))$ with $\beta =1$ (respectively $\beta =1/2$) for the complex (resp. real) Ginibre ensemble. For any $p\in [0,1]$, the equilibrium spectral densities as well as the rate function ${\psi }_R\left(p\right)$ are explicitly derived. This function displays a third order phase transition at the critical (minimum) value $p_R^{*}=1-R^2$, associated to a phase transition of the Coulomb gas. We have deduced that, in the central regime, the fluctuations of the index $N_R$ around its typical value $p_R^{*}N$ scale as $N^{1/3}$.

The heterogeneous gas with singular interaction: Generalized circular law and heterogeneous renormalized energy

Participants : Luis-Carlos Garcia Del Molino, Khashayar Pakdaman [Institut Jacques Monod] , Jonathan Touboul.

We have introduced and analyzed $d$ dimensional Coulomb gases with random charge distribution and general external confining potential [23] . Our long term motivation is to understand the spectrum of random matrices with non identical distributions, for instance with independent elements with distinct statistics. We have shown that these gases satisfy a large deviation principle. The analysis of the minima of the rate function (which is the leading term of the energy) reveals that at equilibrium, the particle distribution is a generalized circular law (i.e. with spherical support but non-necessarily uniform distribution). In the classical electrostatic external potential, there are infinitely many minimizers of the rate function. The most likely macroscopic configuration is a disordered distribution in which particles are uniformly distributed (for $d=2$, the circular law), and charges are independent of the positions of the particles. General charge-dependent confining potentials unfold this degenerate situation: in contrast, the particle density is not uniform, and particles spontaneously organize according to their charge. In that picture the classical electrostatic potential appears as a transition at which order is lost. Sub-leading terms of the energy are derived: we show that these are related to an operator, generalizing the Coulomb renormalized energy, which incorporates the heterogeneous nature of the charges. This heterogeneous renormalized energy informs us about the microscopic arrangements of the particles, which are non-standard, strongly depending on the charges, and include progressive and irregular lattices.