## Section: New Results

### Stochastic Models of Biological Systems

#### Stochastic models for spike-timing dependent plasticity

#####
**Ph. Robert and G. Vignoud**

Synaptic plasticity is a common mechanism used to model learning in stochastic neural networks, STDP is a great example of such mechanisms. We develop a simple framework composed by two neurons and one synaptic weight, seen as stochastic processes and study the existence and stability of such distributions, for a wide range of classical synaptic plasticity models. Using two simple examples of STDP, the calcium-based rule and the all-to-all pair-based rule, we apply stochastic averaging principles and obtain differential equations for the limit processes, based on the invariant distributions of the fast system when the slow variables are considered fixed. We study a general stochastic queue to approximate the calcium-based rule and are able to have an analytical solution for the invariant distribution of the fast synaptic processes. We also detail some simpler systems, either through some approximations or simulations to put into light the influences of different biologically-linked parameters on the dynamics of the synaptic weight.

#### Online Sequence Learning In The Striatum With Anti-Hebbian Spike-Timing-Dependent Plasticity

#####
**G. Vignoud. Collaboration with J. Touboul (Brandeis University)**

Spike-Timing Dependent Plasticity (STDP) in the striatum is viewed as a substrate for procedural learning. Striatal projecting neurons (SPNs) express anti-Hebbian plasticity at corticostriatal synapses, (a presynaptic cortical spike followed by a postsynaptic striatal spike leads to the weakening of the connection, whereas the reverse pairing leads to potentiation ). SPNs need to integrate many inputs to spike, and as such, their main role is to integrate context elements to choose between different sensorimotor associations. In this work, we develop a simple numerical model of the striatum, integrating cortical spiking inputs to study the role of anti-Hebbian STDP in pattern recognition and sequence learning. Cortical neurons are seen as binary input neurons and one striatal SPN is modeled as a leaky-integrate-and-fire neuron. Combined informations from the output, reward and timing between the different spikes modify the intensity of each connection, through two mechanisms: anti-Hebbian STDP and dopaminergic signaling, using three-factor learning rules. We have added a second output neuron with collateral inhibition which leads to an improvement of the global accuracy. In another project, we studied the dynamics of learning, by shutting off/on the dopaminergic plasticity, and compare it to DMS/DLS experimental and behavioral experiments. We show that anti-Hebbian STDP favors the learning of complete sequence of spikes, such as is needed in the striatum, whereas, even if Hebbian STDP helps to correlate the spiking of two connected neurons, it is not sufficient to integrate of long sequences of correlated inputs spikes.

#### D1/D2 detection from action-potential properties using machine learning approach in the dorsal striatum

#####
**G. Vignoud. Collaboration, with Team Venance (CIRB/Collège de France)**

Striatal medium spiny neurons (MSNs) are segregated into two subpopulations, the D1 receptor-expressing MSNs (the direct striatonigral pathway) and the D2 receptorexpressing MSNs (the indirect striatopallidal pathway). The fundamental role of MSNs as output neurons of the striatum, and the necessary distinction between D1- and D2-expressing neurons accentuate the need to clearly distinguish both subpopulations in electrophysiological recordings in vitro and in vivo. Currently, fluorescent labelling of the dopaminergic receptors in mice enables a clear differentiation. However, multiplying in vivo the number of genetic markers (optogenetics, fluorescence) hinders possibilities for other genetic manipulations. Moreover, electrophysiological properties of fluorescents neurons can slightly differ from “native” cells and false-positive can be observed. The lack of a proper way to separate D1- and D2-MSNs based on electrophysiological properties led us to devise a detection algorithm based on action potential profile. We used more than 450 D1/D2 labelled MSNs from in vitro patch-clamp recordings (different experimentalists, different setups and protocols), to characterize and identify properties that facilitate the MSN discrimination. After analyzing passive and active MSN membrane properties, we built an extensive dataset and fed it into classical machine learning classification methods. The training of the different algorithms (k-nearest neighbors, random forest, deep neural networks, …) was performed with the scikit-learn Python library, and the optimized classifier was able to correctly discriminate neurons in the dorsolateral striatum at 76% (and up to 83% if we allow the classifier to reject some MSNs). This study developed an efficient classification algorithm for D1/D2-MSNs, facilitating cell discrimination without specific genetic fluorescent labelling, leaving some room for other genetic markers and optogenetic labeling.

#### The Stability of Non-Linear Hawkes Processes

#####
**Ph. Robert and G. Vignoud**

We have investigated the asymptotic properties of self-interacting point processes introduced by Kerstan (1964) and Hawkes and Oakes (1974). These point processes have the property that the intensity at some point $t\in (-\infty ,+\infty )$ is a functional of all points of the point process before $t$. Such a process is said to be stable if it has a version whose distribution is invariant by translation. By using techniques of coupling and Markovian methods, we have been able to obtain some existence and uniqueness results with weaker conditions than in the current literature.

#### Mathematical Models of Gene Expression

#####
**Ph. Robert**

In Robert [30] we analyze the equilibrium properties of a large class of stochastic processes describing the fundamental biological process within bacterial cells, *the production process of proteins*. Stochastic models classically used in this context to describe the time evolution of the numbers of mRNAs and proteins are presented and discussed. An extension of these models, which includes elongation phases of mRNAs and proteins, is introduced. A convergence result to equilibrium for the process associated to the number of proteins and mRNAs is proved and a representation of this equilibrium as a functional of a Poisson process in an extended state space is obtained. Explicit expressions for the first two moments of the number of mRNAs and proteins at equilibrium are derived, generalizing some classical formulas. Approximations used in the biological literature for the equilibrium distribution of the number of proteins are discussed and investigated in the light of these results. Several convergence results for the distribution of the number of proteins at equilibrium are in particular obtained under different scaling assumptions.

#### Stochastic modelling of molecular motors

##### Marie Doumic, Dietmar Oelz, Alex Mogilner

It is often assumed in biophysical studies that when multiple identical molecular motors interact with two parallel microtubules, the microtubules will be crosslinked and locked together. The aim of the article [4] is to examine this assumption mathematically. We model the forces and movements generated by motors with a time-continuous Markov process and find that, counter-intuitively, a tug-of-war results from opposing actions of identical motors bound to different microtubules. The model shows that many motors bound to the same microtubule generate a great force applied to a smaller number of motors bound to another microtubule, which increases detachment rate for the motors in minority, stabilizing the directional sliding. However, stochastic effects cause occasional changes of the sliding direction, which has a profound effect on the character of the long-term microtubule motility, making it effectively diffusion-like. Here, we estimate the time between the rare events of switching direction and use them to estimate the effective diffusion coefficient for the microtubule pair. Our main result is that parallel microtubules interacting with multiple identical motors are not locked together, but rather slide bidirectionally. We find explicit formulae for the time between directional switching for various motor numbers.