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Section COMPUTATIONAL NEUROSCIENCE
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with the possibility to apply the obtained knowledge for creation of the brain-computer interfaces. It should be noted that a number of research teams and private companies (for example, Google and Honda) are currently working on solution of this complex interdisciplinary problem. However, for realization of such ambitious problem it is necessary to understand the main fundamental processes occurring in the brain during the solution of different tasks. One of such tasks is the problem of cognitive behavior of living subject in the real world. Such function is known to be controlled by the neural activity in the hypothalamus in the brain of mammals. Thus, there is an interesting question related to the study of oscillatory activity of neural ensembles in the hypothalamus by means of the fundamental approaches of nonlinear dynamics.
In present Report we have studied the behavior of rodents being in the state of rest (under the influence of general anesthesia). We have considered the electrical activity observed in the left and right hippocampus of rats using the continuous wavelet transform with the complex basis [3 - 5]. Using continuous wavelet transform spectral analysis was carried activity of local field potentials generators in the left and right hippocampus of rats. The electrical activity observed in the left and right hippocampus of rats, we can distinguish two characteristic modes of behavior primarily is a mode with a slowly varying amplitude oscillations (4 - 12 Hz), the so-called hippocampal the theta rhythm. In addition, you can select the second mode for typical generators field potentials in the left and right hippocampus of rats, this behavior is rapidly changing the amplitude of the oscillations (30 - 60 Hz). Thus, these results confirm that the bond between the generator field potentials in right and left hippocampus of rats is performed in the frequency range (0 - 60 Hz). It should be noted that the degree of coherence (or, in turn, the relationship between the potentials of the field generators in the right and left side of the rat hippocampus) vary depending on the experiments.
We have found the characteristic features of the brain activity in the case when the animal does not solve any problem of cognitive navigation. The study intermitentnoy synchronization fluctuations in the ways Schaffer, in which has been developed and used a new technique based on a previously proposed methods of analysis on different time scales synchronization [6]. Thus it was obtained the dependence of duration synchronous behavior between field potentials generators right and left parts of the rodent hippocampus, which is close to exponential.
References
1. G. Buzsaki, A. Draguhn Neuronal Oscillations in Cortical Networks // Science. 2004. V. 304. P. 1926
2. M. I. Rabinovich et al. Dynamical principles in neuroscience // Rev. Mod. Phys. 78 1213 (2006)
3. B. Torresani, Continuous Wavelet Transform, Savoire, Paris, 1995
4. A. E. Hramov, A. A. Koronovskii, An approach to chaotic synchronization. // Chaos 14 (3), 603 (2004)
5. A. E. Hramov, A. A. Koronovskii, Time scale synchronization of chaotic oscillators. // Physica D 206 (3-4), 252 (2005)
6. Zhuravlev M.O., Koronovskii A.A., Moskaleriko O.I., Ovchinnikov A.A., Hramov A.E. Ring intermittency near the boundary of the synchronous time scales of chaotic oscillators. // Phys. Rev. E. 83, (2011) 027201
Learning in Coupled Neural Networks with Heteroclinic Circuits
A.O. Selskii1 *and V.A. Makarov1'2
1 N.I. Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia;
2 Instituto de Matematica Interdisciplinar, Universidad Complutense de Madrid, Madrid, Spain.
Relatively recently it has been shown that regular but complex enough oscillatory activity in dynamical systems can emerge from the so-called stable heteroclinic channels [1,2]. In a neural network consisting of several coupled cells, one may observe a situation when all neurons are excited sequentially, i.e. each neuron becomes a winner for a limited time. Such a dynamic regime, called winner-less competition (WLC), can be implemented in a vicinity of heteroclinic trajectories connecting saddle equilibria in a loop [3,4]. From the one side, earlier it has been shown that a heteroclinic circuit may exist if certain relationships among synaptic coupling strength in the neural network are fulfilled [5,6]. From the other side, in neuronal systems synaptic plasticity may potentially change dynamic regimes. The latter may enable the emergence of WLC under special network training.
In this work we propose a model of learning, i.e., a learning rule, which allows one neural network, call a teacher, to impose its own dynamic to another neural network, call a learner. As a result, in the learner there appear WLC oscillations synchronized in phase with the oscillations of the teacher.
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Section COMPUTATIONAL NEUROSCIENCE
The model of a neural network that we used is:
dA ,(Q dt
= ^(0 -
[i- I
J+ mo
(1)
were Ai (t) describes the activity of the i-neuron at time instant t, ^ (t) is the Gaussian white noise, and Pj is the matrix of couplings that determines the system dynamic. Figure 1 (left column) shows an example of oscillations in two neural networks (teacher and learner) with different directions of excitation and the period of dominant activities.
For network training we apply the follow learning rule [7]:
*'< = iXtT ciAi )2-A )2 (i-a) ^=a + a
I I
A
A2)
(2)
were c is the learning rate, ai is the element in the coupling matrix of the learner (see [5], all Pi = 2.8 in both system). The elements in the coupling matrix of the teacher are fixed: a1t a\ = 0.2 . At t = 0 we set the elements in coupling matrix of the learner to: a\ = 0.35 a2 = 0.15 a3 = 0.2
0.7, a2 = 0.4
At the beginning both networks exhibit significantly different WLC dynamics (Fig. 1, left column). The teacher implements red->blue->green->red cycle, while the learner follows red->green->blue->red sequence. Then the purpose of learning is to synchronize oscillations in the learner with the teacher by tuning the coupling strengths in the learner. After a transient time the learner drastically changes its dynamics and starts reproducing the teacher cycle (Fig. 1, right column). Thus, we obtained the structural synchronization of two neuronal circuits by learning.
Fig.1. The dynamic of the teacher and learner networks before learning (left column) and after learning (right column)
We note that the learning was implemented through a rule that changes local couplings in the learner according to the own and the teacher dynamics. Thus, no direct influence of the teacher to the state variables of the learner exists. Such a mechanism of synchronization differs significantly from much more common models of synchronization based on couplings among state variables describing the system dynamics and better describes learning among spatially segregated neural networks.
Acknowledgements
This work has been supported by the Russian Science Foundation (project 15-12-10018).
Section COMPUTATIONAL NEUROSCIENCE References
1. Ashwin, P. and Chossat, P., J. Nonlinear Sci., 1998, 8 (2), 103-129.
2. Ashwin, P. and Field, M., Arch. Ration. Mech. Anal., 1999, 148 (2), 107-143.
3. Rabinovich, M.I., Volkovskii, A., Lecanda, P., Huerta, R., Abarbanel, H.D.I., and Laurent, G., Phys. Rev. Lett., 2001, 87 (6), 068102, 4.
4. Varona, P., Rabinovich, M. I., Selverston, A.I., and Arshavsky, Y.I., Chaos, 2002, 12, (3), 672-677.
5. Afraimovich, V. S., Rabinovich, M. I., and Varona, P., Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2004, 14 (4), 1195-1208.
6. Rabinovich, M. I., and Muezzinoglu, M. K., Phys. Uspekhi, 2010, 53 (4), 357-372.
7. Selskii, A., and Makarov, V.A., Regular and Chaotic Dynamics, 2016, 21 (1), 97-106.
Simulations of Chloride Pathology as a Mechanism for Generation of Abnormal Neural Activity
Boris Gutkin*
NRU Higher School of Economics, Center for Cognition and Decision Making, Moscow, Russia. * Presenting e-mail: [email protected]
Pharmacoresistant epilepsy is a chronic neurological condition in which a basal brain hyper excitability results in paroxysmal hyper synchronous neuronal discharges. Human temporal lobe epilepsy has been associated with dysfunction or loss of the potassium-chloride co-transporter KCC2 in a subset of pyramidal cells in the subiculum, a key structure generating epileptic activities. KCC2 regulates intra-neuronal chloride and extracellular potassium levels by extruding both ions. Absence of effective KCC2 may alter dynamics of chloride and potassium levels during repeated activation of GABAergic synapses due to interneuron activity. In turn such GABAergic stress may itself affect Cl- regulation. Such changes in ionic homeostasis may switch GABAergic signaling from inhibitory to excitatory in affected pyramidal cells and also increase neuronal excitability. Possibly they contribute to periodic bursting in pyramidal cells, an essential component in the onset of ictal epileptic events. We tested this hypothesis with a computational model of a subicular network with realistic connectivity. The pyramidal cell model explicitly incorporated the cotransporter KCC2 and its effects on the internal/external chloride and potassium levels Our network model suggested the loss of KCC2 in a critical number of pyramidal cells increased external potassium and intracellular chloride concentrations leading to seizure-like field potential oscillations. These oscillations included transient discharges leading to ictal-like field events with frequency spectra as in vitro. Restoration of KCC2 function suppressed seizure activity and thus may present a useful therapeutic option. These simulations therefore suggest that a reduced KCC2 cotransporter activity alone may underlie the generation of ictal discharges.
This work was supported by the Russian Ministry of Education (Contract 14.6008.21.0001, unique ID project RFME-FI60815X0001).
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The work of the International conference Computational Neuroscience was supported by the RFBR fund (grant 16-01-20459\16).
