B. Cessac, "A discrete time neural network model with spiking neurons II. Dynamics with noise", submitted to J. Math. Biol.

We provide rigorous and exact results characterizing the statistics of spike trains in a network of leaky integrate and fire neurons, where time is discrete and where neurons are submitted to noise, without restriction on the synaptic weights. We show the existence and uniqueness of an invariant measure of Gibbs type and discuss its properties. We also discuss Markovian approximations and relate them to the approaches currently used in computational neuroscience to analyse experimental spike trains statistics.

B. Cessac, H. Paugam-Moisy, T. Viéville, "Overview of facts and issues about neural coding by spike", J. Physiol., Paris, 104, (1-2), 5-18, (2010).

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.

B. Cessac, H. Rostro, J.C. Vasquez, T. Viéville, "How Gibbs distributions may naturally arise from synaptic adaptation mechanisms", J. Stat. Phys,136, (3), 565-602 (2009).

This paper addresses two questions in the context of neuronal networks dynamics, using methods from dynamical systems theory and statistical physics: (i) How to characterize the statistical properties of sequences of action potentials ("spike trains") produced by neuronal networks ? and; (ii) what are the effects of synaptic plasticity on these statistics ? We introduce a framework in which spike trains are associated to a coding of membrane potential trajectories, and actually, constitute a symbolic coding in important explicit examples (the so-called gIF models). On this basis, we use the thermodynamic formalism from ergodic theory to show how Gibbs distributions are natural probability measures to describe the statistics of spike trains, given the empirical averages of prescribed quantities. As a second result, we show that Gibbs distributions naturally arise when considering "slow" synaptic plasticity rules where the characteristic time for synapse adaptation is quite longer than the characteristic time for neurons dynamics.

B. Cessac, H. Rostro, J.C. Vasquez, T. Viéville, "Statistics of spikes trains, synaptic plasticity and Gibbs distributions", NeuroComp 2008 (communication).

We introduce a mathematical framework where the statistics of spikes trains, produced by neural networks evolving under synaptic plasticity, can be analysed.

Slides

B. Cessac, H. Rostro, J.C. Vasquez, T. Viéville, "To which extend is the ``neural code'' a metric ?", NeuroComp 2008 (poster).

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.

J.C. Vasquez, B. Cessac, “PHASE SPACE STRUCTURE OF SPIKING NEURAL NETWORK WITH LAPLACIAN COUPLING”, ARIANE08 Conference, Greece.

Abstract.

Poster.

B. Cessac, T. Viéville, "On Dynamics of Integrate-and-Fire Neural Networks with Conductance Based Synapses", "Frontiers in Neuroscience",July 2008, Volume2, Article 2.

We present a mathematical analysis of a networks with Integrate-and-Fire neurons and adaptive conductances. Taking into account the realistic fact that the spike time is only known within some finite precision, we propose a model where spikes are effective at times multiple of a characteristic time scale d, where d can be arbitrary small (in particular, well beyond the numerical precision). We make a complete mathematical characterization of the model-dynamics and obtain the following results. The asymptotic dynamics is composed by finitely many stable periodic orbits, whose number and period can be arbitrary large and can diverge in a region of the synaptic weights space, traditionally called the ``edge of chaos'', a notion mathematically well defined in the present paper. Furthermore, except at the edge of chaos, there is a one-to-one correspondence between the membrane potential trajectories and the raster plot. This shows that the neural code is entirely ``in the spikes'' in this case. As a key tool, we introduce an order parameter, easy to compute numerically, and closely related to a natural notion of entropy, providing a relevant characterization of the computational capabilities of the network. This allows us to compare the computational capabilities of leaky and Integrate-and-Fire models and conductance based models. The present study considers networks with constant input, and without time-dependent plasticity, but the framework has been designed for both extensions.

B. Cessac. A discrete time neural network model with spiking neurons. Rigorous results on the spontaneous dynamics. Journal of Mathematical Biology, Volume 56, Number 3, 311-345 (2008).

We derive rigorous results describing the asymptotic dynamics of a discrete time model of spiking neurons introduced in [Soula, Beslon, Mazet, Neural Computation 18:1 (2006)]. Using symbolic dynamic techniques we show how the dynamics of membrane potential has a one to one correspondence with sequences of spikes patterns (``raster plots’’). Moreover, though the dynamics is generically periodic, it has a weak form of initial conditions sensitivity due to the presence of a sharp threshold in the model definition. As a consequence, the model exhibits a dynamical regime indistinguishable from chaos in numerical experiments.