arXiv subjects

Gaetan Vignoud

Publications and source records attributed to Gaetan Vignoud.

4 recordsLinked to original sources

arXiv2021

Stochastic Models of Neural Plasticity: A Scaling Approach

In neuroscience, synaptic plasticity refers to the set of mechanisms driving the dynamics of neuronal connections, called synapses and represented by a scalar value, the synaptic weight. A Spike-Timing Dependent Plasticity (STDP) rule is a biologically-based model representing the time evolution of the synaptic weight as a functional of the past spiking activity of adjacent neurons. A general mathematical framework has been introduced in~arXiv:2010.08195. In this paper we develop and investigate a scaling approach of these models based on several biological assumptions. Experiments show that long-term synaptic plasticity evolves on a much slower timescale than the cellular mechanisms driving the activity of neuronal cells, like their spiking activity or the concentration of various chemical components created/suppressed by this spiking activity. For this reason, a scaled version of the stochastic model of~arXiv:2010.08195 is introduced and a limit theorem, an averaging principle, is stated for a large class of plasticity kernels. A companion paper~arXiv:2010.08790 is entirely devoted to the tightness properties used to prove these convergence results. These averaging principles are used to study two important STDP models: pair-based rules and calcium-based rules. Our results are compared with the approximations of neuroscience STDP models. A class of discrete models of STDP rules is also investigated for the analytical tractability of its limiting dynamical system.

math.PR↗

arXiv2021

Stochastic Models of Neural Synaptic Plasticity

In neuroscience, learning and memory are usually associated to long-term changes of neuronal connectivity. In this context, synaptic plasticity refers to the set of mechanisms driving the dynamics of neuronal connections, called {\em synapses} and represented by a scalar value, the synaptic weight. Spike-Timing Dependent Plasticity (STDP) is a biologically-based model representing the time evolution of the synaptic weight as a functional of the past spiking activity of adjacent neurons. If numerous models of neuronal cells have been proposed in the mathematical literature, few of them include a variable for the time-varying strength of the connection. A new, general, mathematical framework is introduced to study synaptic plasticity associated to different STDP rules. The system composed of two neurons connected by a single synapse is investigated and a stochastic process describing its dynamical behavior is presented and analyzed. The notion of plasticity kernel is introduced as a key component of plastic neural networks models, generalizing a notion used for pair-based models. We show that a large number of STDP rules from neuroscience and physics can be represented by this formalism. Several aspects of these models are discussed and compared to canonical models of computational neuroscience. An important sub-class of plasticity kernels with a Markovian formulation is also defined and investigated. In these models, the time evolution of cellular processes such as the neuronal membrane potential and the concentrations of chemical components created/suppressed by spiking activity has the Markov property.

math.PR↗

arXiv2020

Averaging Principles for Markovian Models of Plasticity

Mathematical models of biological neural networks are associated to a rich and complex class of stochastic processes. In this paper, we consider a simple {\em plastic} neural network whose {\em connectivity/synaptic strength} $(W(t))$ depends on a set of activity-dependent processes to model {\em synaptic plasticity}, a well-studied mechanism from neuroscience. A general class of stochastic models has been introduced in \cite{robert_mathematical_2020} to study the stochastic process $(W(t))$. It has been observed experimentally that its dynamics occur on much slower timescale than that of the main cellular processes. The purpose of this paper is to establish limit theorems for the distribution of $(W(t))$ with respect to the fast timescale of neuronal processes. The central result of the paper is an averaging principle for the stochastic process $(W(t))$. Mathematically, the key variable is the point process whose jumps occur at the instants of neuronal spikes. A thorough analysis of several of its unbounded additive functionals is achieved in the slow-fast limit. Additionally, technical results on interacting shot-noise processes are developed and used in the general proof of the averaging principle.

math.PR↗

arXiv2019

Ghost Units Yield Biologically Plausible Backprop in Deep Neural Networks

In the past few years, deep learning has transformed artificial intelligence research and led to impressive performance in various difficult tasks. However, it is still unclear how the brain can perform credit assignment across many areas as efficiently as backpropagation does in deep neural networks. In this paper, we introduce a model that relies on a new role for a neuronal inhibitory machinery, referred to as ghost units. By cancelling the feedback coming from the upper layer when no target signal is provided to the top layer, the ghost units enables the network to backpropagate errors and do efficient credit assignment in deep structures. While considering one-compartment neurons and requiring very few biological assumptions, it is able to approximate the error gradient and achieve good performance on classification tasks. Error backpropagation occurs through the recurrent dynamics of the network and thanks to biologically plausible local learning rules. In particular, it does not require separate feedforward and feedback circuits. Different mechanisms for cancelling the feedback were studied, ranging from complete duplication of the connectivity by long term processes to online replication of the feedback activity. This reduced system combines the essential elements to have a working biologically abstracted analogue of backpropagation with a simple formulation and proofs of the associated results. Therefore, this model is a step towards understanding how learning and memory are implemented in cortical multilayer structures, but it also raises interesting perspectives for neuromorphic hardware.

q-bio.NC↗