Transient sequences in adaptive spiking networks: hypernetworks and spatiotemporal processing V.I. Nekorkin Institute of Applied Physics RAS Nizhny Novgorod
Plan 1) Transient sequential dynamics in neural networks; 2) Cognitive dynamics are competitive and transient — stable heteroclinic channel; 3) Hypernetworks for describing transient sequential dynamics; 4) Example of model implementation; 5) Map-based model of neurons and links; 6) Hypersimplexes and hypernetwork; 7) Spatiotemporal processing; 8) Biological neural networks and hypernetworks; 9) Conclusion; 10) Topical problems and publications.
Spatiotemporal sequential activity in neural networks Many neurophysiologic experiments have indicated that some neural processes (for example, processes related with performing of different cognitive tasks - memory, attention, psychomotor coordination and so on) are accompanied only by transient activity on the level of individual neurons or small enough groups of neurons. In the result of such processes a certain sequence of transitional activity phases appears in neural network. It is clear that such activity of neural networks cannot be understood within the framework of classical models of nonlinear dynamics which are based on concept of attractor because here the main effect is achieved long before the system reaches its neighborhood. Examples of sequential neural activity ✓ olfactory system ✓ gustatory cortex ✓ vocal tract of songbirds Transient patterns in the olfactory system of locust The responses of two simultaneously recorded projection neurons (PN) to the odor cherry (21 trials). M. Wehr, G. Laurent, Odor encoding by temporal sequences of firing in oscillating neural assemblies, Nature, 1996.
Transient dynamics for odor encoding Responses of locust projection neurons to 2 different olfactory stimuli. Three-dimensional graphs were obtained from an analysis of the activity of 87 neurons using dimension reduction methods (including local embedding). Buonomano, D. V., & Maass, W. (2009). State-dependent computations: spatiotemporal processing in cortical networks. Nature Reviews Neuroscience , 10 (2), 113. Broome, B. M., Jayaraman, V., & Laurent, G. (2006). Encoding and decoding of overlapping odor sequences. Neuron , 51 (4), 467-482.
Cognitive dynamics are competitive and transient — stable heteroclinic channel
Winnerless competition is the dynamial origin of a stable heteroclinic chain — sequence of metastable states
Hypersimplexes and hypernetworks Hypersimplexes are ordered sets of vertices with an explicit n-ary relation. Hypernetworks are sets of hypersimplexes. J. Johnson. Hypernetworks in the science of complex systems. World Scientific, 2013.
Model for transient dynamics: traffic in hypernetwork Spatiotemporal processing in a hypernetwork: transient sequences in response to stimuli Hypernetwork: structured set of hypersimplices Family of hypersimplixes: possible activity configurations Artificial neural network Stimulus Maslennikov, O. V., Shchapin, D. S., & Nekorkin, V. I. (2017). Transient sequences in a hypernetwork generated by an adaptive network of spiking neurons. Phil. Trans. R. Soc. A , 375 (2096), 20160288.
Map-based model of neural activity x – membrane potential of the neuron; y – outward ionic currents (recovery variable); J – controls the level of depolarization; ε – time scale of the recovery variable; β, d – control threshold properties of bursting oscillations; a – controls shape of neural impulses. В.И. Некоркин, Л.В. Вдовин. Изв. вузов ПНД. 2007; M.Courbage, V.I. Nekorkin, L.V. Vdovin. Chaos. 2007.
Slow-fast dynamics Fast dynamics ( y is constant) For fixed y there are two different regimes depending on y : Slow dynamics − − − + = F ( x ) H ( x d ) y I 0 , n n n = + − y y ( x J ), + n 1 n n
Gallery of dynamical regimes in map-based model Regular regimes Chaotic regimes
System`s response to stimulus When stimulated by rectangular impulse, the system generates a burst.
Interneuron links where g defines the coupling strength, ν is the reversal potential, and θ is the threshold parameter. Inhibitory coupling: ν = -0.5, θ = 0.2, and g =0.15. Adjacency matrix if the j-th node affects the i-th node at the moment of n otherwise As a result, the network with a constant topology generates a cyclic sequence of clusters, the so-called cluster state
Chaotic oscillatory dynamics a =0.1, β=0, d=0.45, ε=0.01 a =0.1, β=0.3, d=0.45, ε=0.001
Hypersimplex: activity and structure configurations Operator of connection rewiring x i (n) — activity of neuron i E(n) — external stimulus T=T(x i , E, n) — operator of links update due to plastisity G=(G ij ) — matrix of interneuron links G(n+1) = TG(n) – update of links resulting in activation of a new hypersimplex
Hypernetwork Hypernetwork consisting of 30 hypersimplexes describing three- cluster states in the adaptive 5-node spiking network.
Spatial processing Stimulus A No. oscillator time Stimulus B No. oscillator time
Spatiotemporal processing Sk Sequence of activity states No. oscillator time
Spatiotemporal processing S k Sequence of activity states No. oscillator time
Biological neural networks and hypernetworks Courtesy of J. Soriano (University of Barcelona)
Biological neural networks and hypernetworks Fluorescence imaging 1 0 0 1 1 0 1 0 1 0 0 1 ... ... ... ... ... ... ... ... ... 0 0 1 0 Sequence of hypersimplexes Sequence of activity states
Homogeneous network Data: neuron dynamics in vitro Spike rastrogram Number of Sequence of hypersimplexes hypersimplexes observed
Clustered network Spike rastrogram Data: neuron dynamics in vitro Sequence of hypersimplexes Number of hypersimplexes observed
Conclusion • A model was proposed for a spiking neuron network that generates two-level dynamics. At the first level, transient sequences of activity clusters are formed; at the second one — at the hypernetwork — paths of activations between different hypersimplexes are created. • The approach was applied to analysing neurobiological in vitro data and based on the network spiking dynamics, the hypersimplexes were found and the activation paths in the hypernetwork were obtained.
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