Nonlinear dynamics emerging in large scale neural networks with ontogenetic and epigenetic processes J. Iglesias, O.K. Chibirova, A.E.P. Villa Grenoble Institut des Neurosciences-GIN, Centre de Recherche Inserm U 836-UJF-CEA-CHU NeuroHeuristic Research Group, University Joseph Fourier, Grenoble, France {Javier.Iglesias, Olga.Chibirova, Alessandro.Villa}@ujf-grenoble.fr
Neural Network Model Internal Modification External influences Initial Rules Connectivity Rules Background STDP Activity Network Early Developmental Stimulation Phase 2
I nitial Connectivity Rules Network Connectivity Network ● 8,000 of excitatory and 2,000 inhibitory units are distributed (Sobol quasi random distribution) on a 100x100 lattice ● Connections between cells are established following a 2D Gaussian probability density Connectivity probability for excitatory (top) and inhibitory (bottom) units 3
Internal Modification Rules STDP STDP Modification Network Early development ● To implement the STDP rule the following real valued function is used: L ji t 1 = L ji t k act S i t M j t − S j t M i t j − presynaptic ,i − postsynapticunit 4
Internal Modification Rules STDP STDP Modification Network Early development ● Synapses are characterized by an activation level which is an integer valued function: A ji t ∈[ A 1 A 2 A 3 A 4 ] ; 5
Internal Modification Rules Early Developmental STDP Network Phase Early Developmen t ● If the firing rate exceeds a threshold value, the unit definitively stops its activity with some probability. ● The early developmental phase takes place during the first 800 milliseconds of the simulation. After that time the STDP phase starts. 6
External influence Spontaneou s Stimulation Network Activity Stimulation ● The duration of each stimulus is 100ms , its rate is 0.5Hz ● Two sets of units labeled A and B are composed of 400 randomly chosen excitatory units ● Each set is divided into 10 groups of 40 units A ={ A 1 , A 2 , ... A 10 } ; B ={ B 1 , B 2 , ... B 1 0 } and receives a strong depolarization each ms of stimulation either in order AB or in order BA : AB: [ A 1 , A 2 , ... A 10 , B 1 , B 2 , ... B 10 ] ; BA: [ B 1 , B 2 , ... B 10 , A 1 , A 2 , ... A 10 ] , 5 times 5 times 5 times 5 times the sequences of AB and BA stimulations are random and equiprobable 7
External influence Backgroun Background Activity d Network Activity Stimulation ● The background activity simulates the input of afferents to the network ● Throughout the simulation, each unit receives an input equivalent to a fixed number of afferents generating a postsynaptic potential of 0.84mV following an independent Poisson process with mean rate of 5 inputs/s 8
Unit State ● A unit fires if its state function S(t) =1. It depends on the membrane potential and the refractory period: S t 1 = H V t − q H t ref − prev spk t ; H x = 0 : x 0 ; 1 :overwise ● External activity, potential leakage and postsynaptic activity of projecting units contribute to the membrane potential: V i t 1 = V rest B t 1 − S i t V i t − V rest k mem ∑ ji t j ● The postsynaptic potential depends on the synapse activation level: ji t 1 = S j t A ji t P [ q i ,q j ] ; q i ,q j ∈[ excitatory ,inhibitory ] 9
Simulations ● Keeping the same rules and the same parameters the simulations were repeated 30 times with different random generator seeds ● Each seed generates a different network connectivity, as well as different stimulation and background activity patterns Seed Initial Connectivity Background Network Activity Stimulation ● These seeds were reused to reproduce the 30 simulations in absence of stimulation 10
Spatiotemporal Firing Patterns ● Spike trains of all units active by time t=T end =100,000ms except units receiving stimulation were scanned for occurrences of firing patterns ● Pattern Grouping Algorithm (PGA) was used for pattern detection [Tetko,Villa 2001] ● Rasters and occurrence times of two detected patterns: a triplet 〈 23E5,23E5,23E5 ; 154 ± 3.5,364 ± 3.0 〉 0 25 50 75 100 Time [s] -100 0 +154 +364 900 Lag [ms] and a quadruplet 〈 B9B , B9B , B9B , B9B ; 68 ± 2.0,556 ± 1.0,655 ± 4.0 〉 0 25 50 75 100 Time [s] -100 0 +68 +556 +655 900 Lag [ms] 11
Distribution of Detected Patterns stimulation ON : PGA was set to detect patterns of 3 and 4 spikes number of simulations 7 6 The majority of detected patterns are 5 4 composed by events of one single unit 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 number of found patterns stimulation ON OFF stimulation OFF : 147 197 total patterns number of simulations 6 5 59 61 4 61 86 59 138 triplets/ quadruplets 3 86 138 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 5 6 multi-unit patterns number of found patterns 12
Intra Pattern Intervals stimulation ON : Distribution of the first (a,c) and a b the second (b,d) intervals of n=319 20 n=319 20 16 16 count [%] triplets: 12 12 num 0 8 8 0 1 2 3 4 5 6 7 8 9 10 11 12 13 4 4 number of found patterns 0 0 200 400 600 800 200 400 600 800 intervals [ms] In case of quadruplets all sub- stimulation OFF : triplets are taken into account: c d 20 n=319 20 1. 3. 4. 2. 16 16 count [%] 12 12 8 8 4 4 0 0 200 400 600 800 200 400 600 800 intervals [ms] 13
Dynamics of Patterns stimulation ON : stimulation OFF : n=5672 200 250 n=7359 200 150 150 count count 100 100 50 50 0 0 10000 30000 50000 70000 90000 10000 30000 50000 70000 90000 time [ms] time [ms] Cumulative histogram of the detected patterns occurrences, bin=2,000ms 30 30 20 20 count count 10 10 0 0 10000 30000 50000 70000 90000 10000 30000 50000 70000 90000 time [ms] time [ms] Histogram of the detected patterns' onsets, bin=2,000ms 14
Nonlinear Dynamics On the return map of all the events of a simulation that were part of a pattern plotted together a possible attractor trajectories are distinguishable. This gives a hint about a possible underlying non linear dynamical system. 15
Thank You for Your Attention Acknowledgments PERPLEXUS FP6 EU Project #034632 GABA FP6 EU Project #043309 http://www.neuroheuristic.org/
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