arXiv subjects

Bruno Cessac

Publications and source records attributed to Bruno Cessac.

26 records · Page 2Linked to original sources

arXiv2010

Stochastic firing rate models

We review a recent approach to the mean-field limits in neural networks that takes into account the stochastic nature of input current and the uncertainty in synaptic coupling. This approach was proved to be a rigorous limit of the network equations in a general setting, and we express here the results in a more customary and simpler framework. We propose a heuristic argument to derive these equations providing a more intuitive understanding of their origin. These equations are characterized by a strong coupling between the different moments of the solutions. We analyse the equations, present an algorithm to simulate the solutions of these mean-field equations, and investigate numerically the equations. In particular, we build a bridge between these equations and Sompolinsky and collaborators approach (1988, 1990), and show how the coupling between the mean and the covariance function deviates from customary approaches.

math.PR↗

arXiv2009

Introducing numerical bounds to improve event-based neural network simulation

Although the spike-trains in neural networks are mainly constrained by the neural dynamics itself, global temporal constraints (refractoriness, time precision, propagation delays, ..) are also to be taken into account. These constraints are revisited in this paper in order to use them in event-based simulation paradigms. We first review these constraints, and discuss their consequences at the simulation level, showing how event-based simulation of time-constrained networks can be simplified in this context: the underlying data-structures are strongly simplified, while event-based and clock-based mechanisms can be easily mixed. These ideas are applied to punctual conductance-based generalized integrate-and-fire neural networks simulation, while spike-response model simulations are also revisited within this framework. As an outcome, a fast minimal complementary alternative with respect to existing simulation event-based methods, with the possibility to simulate interesting neuron models is implemented and experimented.

nlin.AO↗

arXiv2009

Indisputable facts when implementing spiking neuron networks

In this article, our wish is to demystify some aspects of coding with spike-timing, through a simple review of well-understood technical facts regarding spike coding. The goal is to help better understanding to which extend computing and modelling with spiking neuron networks can be biologically plausible and computationally efficient. We intentionally restrict ourselves to a deterministic dynamics, in this review, and we consider that the dynamics of the network is defined by a non-stochastic mapping. This allows us to stay in a rather simple framework and to propose a review with concrete numerical values, results and formula on (i) general time constraints, (ii) links between continuous signals and spike trains, (iii) spiking networks parameter adjustments. When implementing spiking neuron networks, for computational or biological simulation purposes, it is important to take into account the indisputable facts here reviewed. This precaution could prevent from implementing mechanisms meaningless with regards to obvious time constraints, or from introducing spikes artificially, when continuous calculations would be sufficient and simpler. It is also pointed out that implementing a spiking neuron network is finally a simple task, unless complex neural codes are considered.

q-bio.NC↗

arXiv2009

A constructive mean field analysis of multi population neural networks with random synaptic weights and stochastic inputs

We deal with the problem of bridging the gap between two scales in neuronal modeling. At the first (microscopic) scale, neurons are considered individually and their behavior described by stochastic differential equations that govern the time variations of their membrane potentials. They are coupled by synaptic connections acting on their resulting activity, a nonlinear function of their membrane potential. At the second (mesoscopic) scale, interacting populations of neurons are described individually by similar equations. The equations describing the dynamical and the stationary mean field behaviors are considered as functional equations on a set of stochastic processes. Using this new point of view allows us to prove that these equations are well-posed on any finite time interval and to provide a constructive method for effectively computing their unique solution. This method is proved to converge to the unique solution and we characterize its complexity and convergence rate. We also provide partial results for the stationary problem on infinite time intervals. These results shed some new light on such neural mass models as the one of Jansen and Rit \cite{jansen-rit:95}: their dynamics appears as a coarse approximation of the much richer dynamics that emerges from our analysis. Our numerical experiments confirm that the framework we propose and the numerical methods we derive from it provide a new and powerful tool for the exploration of neural behaviors at different scales.

physics.bio-ph↗

arXiv2008

To which extend is the "neural code" a metric ?

Here is proposed a review of the different choices to structure spike trains, using deterministic metrics. Temporal constraints observed in biological or computational spike trains are first taken into account. The relation with existing neural codes (rate coding, rank coding, phase coding, ..) is then discussed. To which extend the "neural code" contained in spike trains is related to a metric appears to be a key point, a generalization of the Victor-Purpura metric family being proposed for temporal constrained causal spike trains

physics.bio-ph↗

arXiv2008

A mathematical analysis of the effects of Hebbian learning rules on the dynamics and structure of discrete-time random recurrent neural networks

We present a mathematical analysis of the effects of Hebbian learning in random recurrent neural networks, with a generic Hebbian learning rule including passive forgetting and different time scales for neuronal activity and learning dynamics. Previous numerical works have reported that Hebbian learning drives the system from chaos to a steady state through a sequence of bifurcations. Here, we interpret these results mathematically and show that these effects, involving a complex coupling between neuronal dynamics and synaptic graph structure, can be analyzed using Jacobian matrices, which introduce both a structural and a dynamical point of view on the neural network evolution. Furthermore, we show that the sensitivity to a learned pattern is maximal when the largest Lyapunov exponent is close to 0. We discuss how neural networks may take advantage of this regime of high functional interest.

nlin.CD↗

arXiv2007

Effects of Hebbian learning on the dynamics and structure of random networks with inhibitory and excitatory neurons

The aim of the present paper is to study the effects of Hebbian learning in random recurrent neural networks with biological connectivity, i.e. sparse connections and separate populations of excitatory and inhibitory neurons. We furthermore consider that the neuron dynamics may occur at a (shorter) time scale than synaptic plasticity and consider the possibility of learning rules with passive forgetting. We show that the application of such Hebbian learning leads to drastic changes in the network dynamics and structure. In particular, the learning rule contracts the norm of the weight matrix and yields a rapid decay of the dynamics complexity and entropy. In other words, the network is rewired by Hebbian learning into a new synaptic structure that emerges with learning on the basis of the correlations that progressively build up between neurons. We also observe that, within this emerging structure, the strongest synapses organize as a small-world network. The second effect of the decay of the weight matrix spectral radius consists in a rapid contraction of the spectral radius of the Jacobian matrix. This drives the system through the ``edge of chaos'' where sensitivity to the input pattern is maximal. Taken together, this scenario is remarkably predicted by theoretical arguments derived from dynamical systems and graph theory.

q-bio.NC↗

arXiv2006

Linear response, susceptibility and resonances in chaotic toy models

We consider simple examples illustrating some new features of the linear response theory developed by Ruelle for dissipative and chaotic systems [{\em J. of Stat. Phys.} {\bf 95} (1999) 393]. In this theory the concepts of linear response, susceptibility and resonance, which are familiar to physicists, have been revisited due to the dynamical contraction of the whole phase space onto attractors. In particular the standard framework of the "fluctuation-dissipation" theorem breaks down and new resonances can show up oustside the powerspectrum. In previous papers we proposed and used new numerical methods to demonstrate the presence of the new resonances predicted by Ruelle in a model of chaotic neural network. In this article we deal with simpler models which can be worked out analytically in order to gain more insights into the genesis of the ``stable'' resonances and their consequences on the linear response of the system. We consider a class of 2-dimensional time-discrete maps describing simple rotator models with a contracting radial dynamics onto the unit circle and a chaotic angular dynamics $θ_{t+1} = 2 θ_t (\mod 2π)$. A generalisation of this system to a network of interconnected rotators is also analysed and related with our previous studies \cite{CS1,CS2}. These models permit us to classify the different types of resonances in the susceptibility and to discuss in particular the relation between the relaxation time of the system to equilibrium with the {\em mixing} time given by the decay of the correlation functions. Also it enables one to propose some general mechanisms responsible for the creation of stable resonances with arbitrary frequencies, widths, and dependency on the pair of perturbed/observed variables.

nlin.CD↗