Neuronal Networks dynamics, synaptic plasticity, spikes train statistics.
(i) To propose "good" models, namely biologically relevant and mathematically well posed ;
(ii) To make the analysis of these model-dynamics, with rigorous and analytical results if possible, and with a good control on numerical simulations ;
(iii) To interpret and extrapolate these results and compare them to real neuronal networks behaviour.
This constitutes a real
challenge since neuronal networks are dynamical
systems with a huge number of degree of freedom and parameters,
multi-scales organisation with complex interactions. For example
neurons dynamics depend on synaptic connexions while synapses
evolve according to the neurons activity. Analysing this interwoven
activity requires the development of new methods. Methods coming from
theoretical physics and applied mathematics can be adapted to produce
efficient analysis tools while providing useful concepts. I am
developing such methods based on statistical physics (mean-field
theory, Gibbs distribution analysis), dynamical systems theory (global
stability, bifurcation analysis, characterisation of chaotic dynamics)
and ergodic theory (symbolic coding, thermodynamic formalism). I
believe that such an analysis is an important step towards the
characterisation of in vitro or in vivo neuronal neworks, from space
scales corresponding to a few neurons to scales characterising e.g.
cortical columns. With my colleagues, we have characterized
the
dynamics of several firing rate and spiking neurons models and studied
neural mass models with several populations that mimics cortical
columns architecture.
Spike train analysis. Neurons activity results from complex and nonlinear mechanisms leading to a wide variety of dynamical behaviours. This activity is revealed by the emission of action potentials or ``spikes''. While the shape of an action potential is essentially always the same for a given neuron,the succession of spikes emitted by this neuron can have a wide variety of patterns (isolated spikes, periodic spiking, bursting, tonic spiking, tonic bursting, ...), depending on physiological parameters, but also on excitations coming either from other neurons or from external inputs. Thus, it seems natural to consider spikes as ``information quanta'' or ``bits'' and to seek the information exchanged by neurons in the structure of spike trains. Doing this, one switches from the description of neurons in terms of membrane potential dynamics, to a description in terms of spikes trains. This point of view is used, in experiments, by the analysis of raster plots, i.e. the activity of a neuron is represented by a mere vertical bar each time this neuron emits a spike. Though this change of description raises many questions, it is commonly admitted in the computational neuroscience community that spike trains constitute a ``neural code''. This raises however other questions. How is ``information'' encoded in a spike train ? How to measure the information content of a spike train ? As a matter of fact, a prior to handle ``information'' in a spike train is the definition of a suitable probability distribution that matches the empirical averages obtained from measures and there is currently a wide debate on the canonical form of these probabilities. We are developing methods for the characterisation of spike trains from empirical data. On one hand, we have recently shown how Gibbs distribution are natural candidate for spike train statistics in Integrate and Fire models subjected to synaptic plasticity mechanisms. On the other hand, we are developing numerical methods for the characterization of Gibbs distribution from experimental data (see our MACACC projects for details).
Mean-field analysis to neuronal
networks. This method,
well known in the
field of
statistical physics and quantum field theory, is used in the field of neural networks dynamics
with the aim of modeling neural activity at scales integrating the
effect of
thousands of neurons. This is of central importance for several
reasons. First, most
imaging techniques are not able to measure individual neuron activity
(``microscopic'' scale), but
are instead measuring
mesoscopic effects
resulting from the activity of several hundreds to several hundreds of
thousands of
neurons. Second, anatomical data recorded in the cortex reveal the
existence
of structures, such as cortical columns
with a diameter of about 50
micrometers
to 1 millimeter, containing of the order of one hundred to
one thousand neuronsbelonging
to a few different
species. In this case,
information processing does not occur at the scale of
individual neurons
but rather corresponds to an activity integrating the
collective dynamics
of many interacting neurons and resulting in a mesoscopic signal. Dynamic mean-field theory
allows to obtain the equations of evolution of the effective mean-field from microscopic dynamics in
several model-examples.
We have obtained them in several examples of discrete time neural
networks with firing rates, and more recently, derived rigourous
results on the mean-field dynamics of models with several populations
allowing the rederivation of classical phenomenological equations for
cortical columns such as Jensen-Ritt's (see our MACACCprojects for details).
Dynamical effects induced by synaptic and intrinsic plasticity. This
collaboration aims to understand how structure of biological neural
networks is conditioning their functional capacities, in particular
learning. On one hand we are analysing the dynamics of neural network
models submitted to Hebbian learning and investigating how the capacity
to recognize objects emerges. What are the effect on dynamics and
topology ? For this we are using concepts coming from random
networks, and non linear analysis (see next item). On the other hand,
we are using the informations obtained via this analysis to construct
artificial neural network architecture able to learn basic objects and
then to perform generalization by emergent dynamics. A typical example
is the architecture of the V1 cortex (vision) that we are using as a
guideline. In the long term, the goal would be to produce new computer
architectures inspired from biological networks.
Interplay
between synaptic graph structure and topology.
Neuronal networks can be regarded as graphs where each neuron is a
vertex and each synaptic connection is an edge. The models have usually
simple topologies (e.g. feed forward or recurrent neural networks) but
recent research on nervous and brain systems suggests that the actual
topology of a real-world neuronal system is much more
complex :
small-world and scale-free properties are for example observed in human
brain networks. There is also a complex interplay between the
topological structure of the synaptic graph and the non linear
evolution of the neurons. Thus, the existence of synapses between a
neuron (A) and another one (B) is implicitly attached to a notion of
``influence’’ or causal and directed action from A to B. However, the
neuron B receives usually synapses from many other neurons, each of
them being ``influenced’’ by many other neurons, possibly acting on A,
etc... Thus, the actual ``influence’’ or action of A on B has to be
considered dynamically and in a global sense, by considering A and B
not as isolated objects, but, instead, as entities embedded in a system
with a complex interwoven dynamical evolution. It is thus necessary to
develop tools allowing to handle this interplay. In this spirit we are
using the linear response approach (see here
for details ). These results could lead to new directions in neural
network analysis and more generally in the analysis of non linear
dynamical systems on graphs. However the results quoted above were
obtained in a specific model example and further investigations must be
done, in a more general setting. In this spirit, the present project
aims to explore two directions. Recurrent model with spiking neurons
(see item 1 above) and Complex architecture and learning (item 2
above).