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Investigation of brain and its functions is one of the most intriguing intellectual tasks of modern science. Leading countries have initiated major brain research programs, the creation of a national neuroscience project in Russia has been also discussed at the state level and is currently under way. In this regard, the attempt to combine the efforts of various scientific disciplines in the search for new approaches to this problem is important and timely. One of the most interesting prospects in this novel area is the view of the brain as a complex physical system accessible to research methods of theoretical physics and mathematics. Taking into account such a point of view, and understanding that brain research is a topic that goes beyond our individual knowledge and skills, we would like to invite scientists working in various fields of theoretical physics, physics of complex systems, and mathematics, to express their views on the potential research directions of the principles of brain structure and functioning and, in a broader context, of higher cognitive functions of the brain.

To initiate the communication, on December 4 and 5, 2019 we are organizing a mini-workshop “**Theoretical physics and mathematics of the brain: bridges across disciplines and applications**”. The first day (Tuesday) will be held in the Moscow State University, the second day (Wednesday) - in Skoltech. The organizers of this event are: **Konstantin Anokhin** (Institute for Advanced Brain Studies of the Moscow State University), **Alexander Gorsky** (IITP RAS), **Yuri Kotelevtsev** (Center for Neurobiology and Brain Restoration of Skoltech) and **Sergei Nechaev** (Center Poncelet).

The tentative list of topics of the workshop include:

**Geometric and dynamic properties of brain networks, spreading of excitations on networks****The brain as a critical system: theoretical and experimental points of view****Information and entropic aspects of brain functioning****Topological data analysis, mapping of a connectome into a space of stimuli****Connectome as a statistical system and its relation to other areas of complex distributed systems**

The speakers at the workshop are:

**Konstantin Anokhin**(Moscow State University, Moscow)**Anton Ayzenberg**(Higher School of Economics, Moscow)**Alexander Bernstein**(Skoltech, Moscow)**Roman Borisyuk**(University of Exeter, UK)**Vsevolod Chernyshev**(Higher School of Economics, Moscow)**Alexander Gorsky**(Institute for Information Transmission Problems, Moscow)**Vladimir Itskov**(Pennsylvania State University, USA)**Yakov Kazanovich**(Institute of Mathematical Problems of Biology, Pushino)**Victor Kazantsev**(Lobachevsky University, Nizhny Novgorod)**Alexey Koulakov**(Cold Spring Harbor Laboratory, USA)**Sergey Lobov**(Lobachevsky University, Nizhny Novgorod)**Vladimir Nekorkin**(Institute of Applied Physics, Nizhny Novgorod)**Alexey Ossadtchi**(Higher School of Economics, Moscow)**Elena Popova**(Skoltech, Moscow)**Nikita Pospelov**(Moscow State University, Moscow)**Vadim Ushakov**(Kurchatov Institute, Moscow)

**Theoretical physics and mathematics of the brains: Bridges across disciplines and applications**

### Arrival and coffee

### Fundamental brain theory: the main challenge to theoretical physics and mathematics of the brain

Despite an impressive progress in neuroscience we still lack satisfactory understanding of higher brain functions. What is currently missing is not just more facts, but rather a fundamental brain theory that should bridge neural and mental phenomena, make sense of already existing data and guide new experiments. I will suggest that for the purposes of such theory brain can be best described as neuronal hypernetwork that is amenable to analysis by tools of theoretical physics and mathematics.

### The brain: a homework for a theoretical physics

The experimental results in neuroscience provide the serious challenges for theoretical physics and mathematics. The several groups of questions shall be discussed in the review talk supplemented with the suitable theoretical physics tools. The first group of questions concerns properties of the structural and functional connectoms. The corresponding tools are provided by the theory of multiplex networks and we shall emphasize the importance of the spectral methods. The next group of questions concerns the proper identification of a reasonable statistical model for the brain description. We shall provide the arguments that the generalization of 3D Ising-like model with higher dimensional target space could serve as the toy model. There are several experimental indications that a brain works in the near-critical regime and we shall discuss the possible theoretical tools to identify such criticality. Finally we shall focus on the possible qualitative measures and order parameter for the state of consciousness.

### Coffee break

### Topological analysis and inferring the sensory stimulus space from neural responses

The brain builds its internal sensory representations based on the structure of neural activity. A number of modalities, such as the early visual system and the spatial map in hippocampus, are relatively well-characterized. However some sensory systems, such as olfaction, remain enigmatic. Here the primary difficulty lies in that the underlying perceptual space is not well-understood. Can we "build" the sensory space from neural activity alone, without a prior understanding of how the stimuli are organized? It turns out that ideas and tools "borrowed" from algebraic topology and geometry are well-suited for answering this question.

I will describe a set of mathematical tools that allow to infer the dimension and some geometrical and topological features of stimulus space and illustrate their utility for two neural systems: hippocampus and early olfaction.

### Modeling brain cognitive functions by oscillatory neural networks

We describe an oscillatory neural network designed as a system of generalized phase oscillators with a central element. It is shown that a winner-take-all principle can be realized in this system in terms of the competition of peripheral oscillators for the synchronization with a central oscillator. Several examples illustrate how this network can be used for the simulation of various cognitive functions: consecutive selection of objects in the image, visual search, and multiple object tracking.

### Lunch

### Approaches to Learning and Creating Brain Models

In the presented report, the main approaches to the study and creation of architectures of the neural networks of the brain that ensure the functioning of cognitive processes will be analyzed. Perspective methods for obtaining and processing neurophysiological data (EEG, fMRI, MEG, etc.) and their use for analysis of brain states will be considered.

### Topological Data Analysis of Connectivity Matrices in Medical Diagnostic Tasks

Diagnosis of neurodegenerative diseases based on neuroimaging data is one of the actively researched areas of data science (including data mining and machine learning). Data analysis methods construct decision rules, based on neuroimaging data (MRI, EEG, etc.) of disease-affected patients and healthy volunteers. By applying these rules to the same type of data from a new subject with an unknown decease status, he can be reliably assigned to a group of diseased or healthy subjects. The result of some neuroimaging technologies (fMRI, EEG, etc.) are time series of signals recorded in various regions of the subject’s brain. Using various methods, connectivity matrices are built out of time series, characterizing functional interrelationships between different brain regions (symmetric correlation matrices of functional connectivity, asymmetric matrices of effective connectivity, reflecting causal relationships, etc.). Next, making use of a set of connectivity matrices of disease-affected and control groups of patients, the diagnostic task is to find such matrices’ characteristics (features, biomarkers) that differ significantly for disease-affected and control groups of patients and allow to construct feature-based decision rules.

The performed research consisted in studying the potential resolving power of various features extracted from (symmetric and asymmetric) connectivity matrices based on neuroimaging data (resting state functional magnetic resonance or electroencephalography) of real patients (with diagnoses of epilepsy, depression, etc.) and healthy volunteers. These matrices can be also considered as the families of connectivity graphs (non-oriented or oriented) constructed using a threshold technique with various threshold values. The biomarkers based on spectral characteristics of connectivity matrices (eigenvalues or some polynomials of eigenvalues of initial matrix, graph Laplacian matrix constructed from it, etc.), standard graph-theoretic features of connectivity graphs, as well various topological invariants of simplicial and cell complexes corresponding to symmetric or asymmetric connectivity matrices, were studied and compared. Carried out research has led to the constructing of new biomarkers, based on various homological invariants (Betti numbers, persistence diagrams, Wasserstein distance on a space of persistence diagrams, etc.), having a significant “resolving power”.

*In collaboration with V. Bukhshtaber, E. Burnaev, M. Sharaev, O. Kachan, E. Streltsova*

### Coffee break

### Topological formal contexts

We are interested in connections between the stimuli space and the activity patterns of place cells. One of the approaches to describe this connection is based on Alexandroff nerve theorem from homotopy theory. We extend this approach by incorporating ideas from formal concept analysis - a well-developed method of data mining. The proposed approach is consistent with topology: under certain assumptions, the homotopy type of the stimuli space can be recovered from the so called "lattice of concepts" formed in an animal's brain.

### Round table discussion

### Arrival

### Transient sequences in adaptive spiking networks: hepernetworks and spatiotemporal processing

We propose a model of an adaptive network of spiking neurons that gives rise to a hypernetwork of its dynamic states at the upper level of description. Left to itself, the network exhibits a sequence of transient clustering which relates to a traffic in the hypernetwork in the form of a random walk. Receiving inputs the system is able to generate reproducible sequences corresponding to stimulus-specific paths in the hypernetwork.

### Structural and functional properties of a nervous system: Modelling tadpole locomotor behaviour in response to sensory signals

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.

### Coffee break

### Formation of neural connectivity: nature versus nurture

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.

### Mapping normal and pathological brain function with magnetoencephalography

Magnetoencephalography(MEG) opens a non-invasive window into the brain function and allow us to explore neural activity on sub-centimeter spatial scale with millisecond temporal resolution. Mathematical methods for analysis of MEG data serve to transform sensor measurements into the cortical activation maps and make MEG a unique brain imaging tool with resolution properties that surpass all other existing non-invasive brain imaging modalities.

I will describe our recently developed methods that yield finer cortical activation maps when analyzing MEG data from cognitive experiments, imaging functional connectivity and exploring abnormal interictal activity in patients with epilepsy.

### Lunch

### Computer simulation of radiation-induced dysfunction of the neural networks of the prefrontal cortex

The study of the effect of radiation on the brain and its cognitive functions is currently an urgent problem. There are natural sources of radiation and artificial, created by man. Natural radiation is cosmic radiation, the dose of which increases with altitude. For flights at high altitudes, their effect must be considered. Man-made sources of radiation require special precautions, such as measures to control the operation of nuclear power plants, precautions when handling radioactive materials, the inevitable dose of radiation when used in medicine (radiation therapy). The aim of our work is a computer simulation of radiation-induced disruption of the neural networks activity of the prefrontal cortex for various options for doses of radiation. To do this, we used a working memory neural network model, which describes neural activity at various input parameters corresponding to different doses of radiation.

(Joint work with A. Bugay and E. Dushanov from LRB JINR, Dubna, Russia)

### Spectral peculiarity and criticality of the human connectome

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.

### Mathematical and computational models of plasticity and learning in spiking neuronal networks (SNN)

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.

### Round table discussion

No preliminary registration is previewed, the participation is free.

To enter the Lomonosov building of the Moscow State University and Skoltech, please consult the page "VENUE".

**On December 4, 2019** the conference is hosted by Institute for Advanced Brain Studies at the Lomonosov building (Ломоносовский корпус) of the Moscow State University, Room **D1** (Lomonosovskiy Prospekt, 27 building 1, Moscow, 119192).

В связи с тем, что проход в МГУ ограничен и осуществляется по предварительной записи, для прохода, пожалуйста, отправьте письмо с Вашими ФИО** на русском языке **на адрес** **ncc.msu@gmail.com **до 2 декабря** и возьмите с собой паспорт. В заголовке письма, пожалуйста, укажите "Конференция 4 декабря". Проход также возможен *без предварительной запси* для всех сотрудников и студентов МГУ, а также **для бывших студентов МГУ** по предъявлении диплома об окончании МГУ и паспорта.

The "Lomonosov building" of MSU is located in 10 min walking distance from the metro station "**Universitet**" along the Lomonosovskii prospect.

**On December 5, 2019** the conference is hosted by the Center for Neurobiology and Brain Restoration of Skolkovo Institute of Science and Technology (Bolshoy Boulevard 30, bld. 1 Moscow, Russia 121205), Room **E-B4-3006 **

To get to the place, please take the bus with the logo **Sk **at the metro station "**Slavyansky boulevard**" (last car from the center) till the stop "Technopark" (inside the Skoltech campus). Then take the local cab No.5 at the same stop till the "University" (big round building). The total trip from the metro takes about 35 min (20 min **Sk** bus + 5 min local cab + 5 min waiting time).

All talks (on Wednesday and Thursday) begin at 10-00 and the round-table discussion is previewed at 17-15.

For any questions please write/call to Sergei Nechaev (sergei.nechaev@gmail.com, +7 916 561 58 72)