Theoretical physics and mathematics of the brain: Bridges across disciplines and applications

Information processing in the brain is based on communication between spiking neurons that are embedded in a network of connections. Although, in most animals, brain connectivity varies between individuals, behaviour is often similar across a species. What fundamental structural properties are shared across individual networks that define this behaviour? We developed a model of pair-wise connectivity in the hatchling Xenopus tadpole nervous system which, when combined with a spiking model of the Hodgkin-Huxley type, reliably produces rhythmic activity corresponding to swimming. We consider three sensory pathways, decision-making neurons and the central pattern generator of swimming. Building the model, we use numerous data to reproduce biologically realistic activity patterns of initiation, continuation, acceleration and termination of swimming.  Model simulations demonstrate how the integration of sensory inputs can produce appropriate motor behaviours mimicking the interaction with external environment.

Neural development leads to the establishment of precise connectivity in
the nervous system. By contrasting the information capacities of cortical connectivity and the genome, I will argue that simplifying rules are necessary in order to create cortical connections from the limited set of
instructions contained in the genome. Such rules contain compact statistical summary of our prior evolutionary experience and form the blueprint for the cognitive capacity of human brain. I will review the mathematical formalism that can explain a wide range of data on the interplay between molecular and experience-dependent mechanisms of connectivity formation.

We have performed the comparative spectral analysis of structural connectomes for various organisms using open-access data. Our analysis indicates several new peculiar features of the human connectome. We found that the spectral density of human connectome has the maximal deviation from the spectral density of the randomized network compared to all other organisms. For many animals except human structural peculiarities of connectomes are well reproduced in the network evolution induced by the preference of 3-cycles formation. To get the reliable fit , we discovered the crucial role of the conservation of local clusterization in human connectome evolution. We investigated for the first time the level spacing distribution in the spectrum of human connectome graph Laplacian. It turns out that the spectral statistics of human connectome corresponds exactly to the critical regime familiar in the condensed matter physics which is hybrid of Wigner-Dyson and Poisson distributions. This observation provides the strong support for the much debated statement of the brain criticality.

Learning in artificial neural networks (ANNs) represents one of the key fundamental stones underlying digital neuroprocessing and artificial intelligence (AI). Based on perceptron-like ANNs technologies of AI are now entering many areas of life. Reflecting “logically” basic cognitive functions, such as learning and memory, the ANNs still stay far from a correct replication of brain functions and true biological “processing”. The reason is that the “algorithms” of brain functioning at the level of brain network circuits are still largely unknown in fundamental science. To understand these “algorithms” biologically detailed models have been developed in computational neuroscience. They can replicate quite rigorously structure and dynamics of neurons and synaptic connectivity. However, it is hardly possible to tune them on solving particular cognitive tasks, for example, to memorize, store and recall information, to implement a motor control or other functions. Spiking neuronal networks (SNN) also representing biologically based models have attracted much attentions in recent years with the hopes to generate real brain inspired technology in silico. The SNNs are much simpler than the biologically detailed (often compartmental) models, hence can be implemented in electronics, but they preserve internal dynamics of neurons and synapses generating spikes and spike discharges of variable shapes. On the one hand, SNNs should be capable with digital processing with spikes, on the other hand, they can realize analogous computations using natural frequencies, phases, synchronization, pattern formation and other phenomena observed in real brain networks. However, it is still unknown weather well defined learning algorithms from ANNs can be translated to SNNs or their brain-inspired dynamics can be tuned to realize particular functions concurrently with existing digital solutions. We present SNNs supplied with spike-timing dependent plasticity (STDP) and consider different models of concurrent learning in the frameworks of temportal and frequency coding. We show how these networks can be adaptively self-organized implementing sensory signal processing and intelligent navigation of roving robots. Based on the concurrent associative learning the SNNs further implement concrete technological tasks of electromyographic (EMG) signal classification and robot control.

This work was supported by the Ministry of Education and Science of Russian Federation under project 14.Y26.31.0022.