Hebbian learning of recurrent connections: a geometrical perspective

Endotaxis: A neuromorphic algorithm for mapping, goal-learning, navigation, and patrolling

An animal entering a new environment typically faces three challenges: explore the space for resources, memorize their locations, and navigate towards those targets as needed. Here we propose a neural algorithm that can solve all these problems and operates reliably in diverse and complex environments. At its core, the mechanism makes use of a behavioral module common to all motile animals, namely the ability to follow an odor to its source. We show how the brain can learn to generate internal "virtual odors" that guide the animal to any location of interest. This endotaxis algorithm can be implemented with a simple 3-layer neural circuit using only biologically realistic structures and learning rules. Several neural components of this scheme are found in brains from insects to humans. Nature may have evolved a general mechanism for search and navigation on the ancient backbone of chemotaxis.

Stability analysis of a neural field self-organizing map

We provide theoretical conditions guaranteeing that a self-organizing map efficiently develops representations of the input space. The study relies on a neural field model of spatiotemporal activity in area 3b of the primary somatosensory cortex. We rely on Lyapunov's theory for neural fields to derive theoretical conditions for stability. We verify the theoretical conditions by numerical experiments. The analysis highlights the key role played by the balance between excitation and inhibition of lateral synaptic coupling and the strength of synaptic gains in the formation and maintenance of self-organizing maps.

Closed-loop stimulation of a delayed neural fields model of parkinsonian STN-GPe network: a theoretical and computational study

Several disorders are related to pathological brain oscillations. In the case of Parkinson's disease, sustained low-frequency oscillations (especially in the β-band, 13-30 Hz) correlate with motor symptoms. It is still under debate whether these oscillations are the cause of parkinsonian motor symptoms. The development of techniques enabling selective disruption of these β-oscillations could contribute to the understanding of the underlying mechanisms, and could be exploited for treatments. A particularly appealing technique is Deep Brain Stimulation (DBS). With clinical electrical DBS, electrical currents are delivered at high frequency to a region made of potentially heterogeneous neurons (the subthalamic nucleus (STN) in the case of Parkinson's disease). Even more appealing is DBS with optogenetics, which is until now a preclinical method using both gene transfer and deep brain light delivery and enabling neuromodulation at the scale of one given neural network. In this work, we rely on delayed neural fields models of STN and the external Globus Pallidus (GPe) to develop, theoretically validate and test in silico a closed-loop stimulation strategy to disrupt these sustained oscillations with optogenetics. First, we rely on tools from control theory to provide theoretical conditions under which sustained oscillations can be attenuated by a closed-loop stimulation proportional to the measured activity of STN. Second, based on this theoretical framework, we show numerically that the proposed closed-loop stimulation efficiently attenuates sustained oscillations, even in the case when the photosensitization effectively affects only 50% of STN neurons. We also show through simulations that oscillations disruption can be achieved when the same light source is used for the whole STN population. We finally test the robustness of the proposed strategy to possible acquisition and processing delays, as well as parameters uncertainty.

Continuous neural network with windowed Hebbian learning

We introduce an extension of the classical neural field equation where the dynamics of the synaptic kernel satisfies the standard Hebbian type of learning (synaptic plasticity). Here, a continuous network in which changes in the weight kernel occurs in a specified time window is considered. A novelty of this model is that it admits synaptic weight decrease as well as the usual weight increase resulting from correlated activity. The resulting equation leads to a delay-type rate model for which the existence and stability of solutions such as the rest state, bumps, and traveling fronts are investigated. Some relations between the length of the time window and the bump width is derived. In addition, the effect of the delay parameter on the stability of solutions is shown. Also numerical simulations for solutions and their stability are presented.

Synchronization in phase-coupled Kuramoto oscillator networks with axonal delay and synaptic plasticity

We explore both analytically and numerically an ensemble of coupled phase oscillators governed by a Kuramoto-type system of differential equations. However, we have included the effects of time delay (due to finite signal-propagation speeds) and network plasticity (via dynamic coupling constants) inspired by the Hebbian learning rule in neuroscience. When time delay and learning effects combine, interesting synchronization phenomena are observed. We investigate the formation of spatiotemporal patterns in both one- and two-dimensional oscillator lattices with periodic boundary conditions and comment on the role of dimensionality.