This paper has been withdrawn. Its main conclusions have been published in On dynamics of integrate-and-fi re neural networks with conductance based synapses, arXiv:0709.4370 and http://www.frontiersin.org/computational_neuroscience/10.3389/neuro.10/002.2008/abstract
This paper has been withdrawn due to an error in the proof of the main theorem.
We consider the evolution of a network of neurons, focusing on the asymptotic behavior of spikes dynamics instead of membrane potential dynamics. The spike response is not sought as a deterministic response in this context, but as a conditional proba
bility : Reading out the code consists of inferring such a probability. This probability is computed from empirical raster plots, by using the framework of thermodynamic formalism in ergodic theory. This gives us a parametric statistical model where the probability has the form of a Gibbs distribution. In this respect, this approach generalizes the seminal and profound work of Schneidman and collaborators. A minimal presentation of the formalism is reviewed here, while a general algorithmic estimation method is proposed yielding fast convergent implementations. It is also made explicit how several spike observables (entropy, rate, synchronizations, correlations) are given in closed-form from the parametric estimation. This paradigm does not only allow us to estimate the spike statistics, given a design choice, but also to compare different models, thus answering comparative questions about the neural code such as : are correlations (or time synchrony or a given set of spike patterns, ..) significant with respect to rate coding only ? A numerical validation of the method is proposed and the perspectives regarding spike-train code analysis are also discussed.
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.
We consider the deterministic evolution of a time-discretized spiking network of neurons with connection weights having delays, modeled as a discretized neural network of the generalized integrate and fire (gIF) type. The purpose is to study a class
of algorithmic methods allowing to calculate the proper parameters to reproduce exactly a given spike train generated by an hidden (unknown) neural network. This standard problem is known as NP-hard when delays are to be calculated. We propose here a reformulation, now expressed as a Linear-Programming (LP) problem, thus allowing to provide an efficient resolution. This allows us to back-engineer a neural network, i.e. to find out, given a set of initial conditions, which parameters (i.e., connection weights in this case), allow to simulate the network spike dynamics. More precisely we make explicit the fact that the back-engineering of a spike train, is a Linear (L) problem if the membrane potentials are observed and a LP problem if only spike times are observed, with a gIF model. Numerical robustness is discussed. We also explain how it is the use of a generalized IF neuron model instead of a leaky IF model that allows us to derive this algorithm. Furthermore, we point out how the L or LP adjustment mechanism is local to each unit and has the same structure as an Hebbian rule. A step further, this paradigm is easily generalizable to the design of input-output spike train transformations. This means that we have a practical method to program a spiking network, i.e. find a set of parameters allowing us to exactly reproduce the network output, given an input. Numerical verifications and illustrations are provided.
We introduce a mathematical framework where the statistics of spikes trains, produced by neural networks evolving under synaptic plasticity, can be analysed.
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 textit{finite} precision, we propose a model where spikes
are effective at times multiple of a characteristic time scale $delta$, where $delta$ can be textit{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.
