Some Recent Study in Biological Neuronal Oscillatory NetworksQingyun Wang(王青云)Department of Dynamics and Control, Beihang University, Beijing 100191Email: qingyunwangbuaa.edu.cn9/26/20241 With complimentsaProf. Lu Qishao(Beihang University)aProf. Duan ZhiSheng (Peking University)aProf. Chen GuanRong (City University of HongKong)aDr. Matjaz Perc (University of Maribor, Koroska cesta 160, SI-2000 Maribor, Slovenia)9/26/20242ContentsFMotivationFNeuronal structure and networkFDynamics of a single neuronFSynchronization transition and Rhythm dynamics on networksFBursting Dynamics on Realistic Neuronal Networks FSpatial Resonance in Neuronal NetworksFOpen problems 9/26/20243The single neuron in the vertebrate cortex connects to more than 10,000 postsynaptic neurons via synapses forming complex networks.The biological nervous system can communicate and transform information through interaction of networks.Neuronal information encoding and processing are reflected by oscillation patterns of electrical impulse sequences in neurons.Motivation9/26/20244Neurodynamics deals with the dynamics and control problems of electrical activities and information processes in nervous systems and is important in the research of biological motions and intelligent behavior. Neurodynamics is a multi-disciplinary research area in nonlinear dynamics, network science and neuroscience.Synchronization is of great importance for signal encoding and transduction in information processing of neurons in two respects.Biological neurons can extract the hindered information by stochastic resonance.9/26/202456 Neural Medicine and Engineering Neuronal firing and information encoding Biological ControlBiological complex neuronal networksIntelligent RobotsBiological structure of nervous systems Neural Cognitive ScienceNeural ComputersNeural Medical Techniques9/26/20246Structure of Single Neuron A basic element for information processing in nervous systems.There are about 1011 neurons in the human brain and 104 synapses for a neuron.9/26/20247Model of neuron (1) (4) (3) (2) Hodgkin A. L., Huxley A. F., J. physiol.(London), 1952, (117) 500-544oHodgkin-Huxley model (1952 Nobel Prize)nChay model, HR model, FHN model, Rulkov map etc.Nonlinear ODEs involving multiple time scales. Hard to analyze both mathematically and computationally.9/26/20248Diversity of firing patterns of a single neuron(in experimental neural pacemakers) (Li Li, Ren Wei, 2003, 2004 ) 9/26/20249Bifurcation scenario (in theoretical model and experimental neural pacemakers)(Li Li, Ren Wei et al, IJBC, 2004. Duan Li Xia et al, Neurocomputing 2008)Chay model9/26/202410Chaos (in experimental neural pacemakers) between period 2 and period 3 firings (Ren Wei, IJBC, 1997) 9/26/2024119/26/202412Neuronal networksoWhat is a Brain?dense networkmore than 104 cell bodies and several kilometers of wires per cubic millimeter.Many types of cells: spiking neurons (1011 cells), glia cells (supporter cells not involved in the information processing) and analog neurons.9/26/202413Synapse and delay9/26/202414Dorsal and lateral views of the connectivity backbone of human brain. Labels indicating anatomical subregions are placed at their respective centers of mass. Nodes (individual ROIs) are coded according to strength and edges are coded according to connection weight .Hagmann P, Cammoun L, Gigandet X, Meuli R, Honey CJ, Wedeen VJ, Sporns O (2008) Mapping the structural core of human cerebral cortex. PLoS Biology Vol. 6, No. 7, e159 9/26/202415Neuronal networksNeuronal networks involve a large number of individual neurons.Details of the connectivity not usually known.Hard to analyze how connectivity influences ODE dynamics.9/26/202416Neuronal synchronizationoSynchronization seems to be very crucial for the efficient processing and transmission of information across the nerve system.oParkinsons tremor is related to neuronal synchronization.oTransiently synchronized assemblies of neurons are believed to underlie cognitive functions.oThere is consistent evidence for a reduction of synchronization in schizophrenia.9/26/202417Robert C. Elson et al, Phys. Review Lett 81 5692-5695 1998Two coupled neuron circuitSynchronized motions9/26/202418Hua Guang GU et al, CHIN. PHYS. LETT. Vol. 26, No. 11 (2009) 110501Experiment results9/26/202419PNAS 2010Listener-speaker can validly communication by means of neuronal coupling9/26/202420Classification of synchronization9/26/202421Burst synchronization and desynchronization in fold/homoclinic (square-wave) bursters. Shown are simulations of coupled Morris-Lecar systems with additional slow variables.Various regimes of synchronization of bursters (modied from Izhikevich, 2000a).9/26/202422Singular phase locking9/26/202423Analysis method of phase lockingoSingular perturbation theory (Terman etal, J Comput Neurosci (2007) 23:217235)Dynamic clustering in the (a) full and (b) reduced models.There are twenty pairs of E- and I-cells. Each E-cell excitesone I-cell and each I-cell inhibits three E-cellsLimited: Network structure, connection style9/26/202424oPhase response curve (discrete map method)Carmen Canavier9/26/202425There are still many other methods: Hilbert transformation of time series, Poincare map method (numerical computation)9/26/202426Incorporating noise and delayoNoise induced (enhanced) synchronization. (two coupled neurons and large scale neuronal networks)oNoise induced spatiotemporal wave patterns (target wave, spiral wave (Ma jun), spatial coherence)oDelay induced complexity in neuronal networks (various synchronization transition and spatial coherence)9/26/202427Effect of delay on two coupled neuronsQ Y Wang et al, IJBC, 18, 4 (2008) 118911989/26/202428Effect of delay on large scale neuronal networksDynamics model9/26/202429Zig-zag frontsAntiphase clustering sunhcronizationIn phase synchronization again9/26/202430Measure:Q.Y. Wang, Z.S Duan, M. Perc, G. R. Chen, EPL, 83 (2008) 500089/26/202431(Continued)A.-L. Barabsi and R. Albert, Science 286, 509 19999/26/202432Increasing delay9/26/2024339/26/2024349/26/2024359/26/2024369/26/202437Qingyun Wang, Matjaz Perc, Zhi Sheng Duan, Guan Rong Chen, Physical Review E, 80, 026206,2009.9/26/202438Attractive coupling9/26/202439Repulsive coupling9/26/202440Bursting periodo 9/26/202441Bursting synchronizationMacaque cortical network, consisting of 95 nodes and 2402 links9/26/202442Mathematical model9/26/202443Bursting synchronizationSpace-time plots of x(t) as obtained for D=2.0 and I_ext=3.2 with different information transmission delays tau, equalling: (a) 0, (b) 7, (c) 14, (d) 22, (e) 30. Notice the bursting synchronization and spikes in burst increase as tau increases.Liang Xiao Ming et al, 2009 PRE9/26/202444In order to quantitatively describe global network states, we define the quantity mThe more synchronous the neuronal network is, the larger is the parameter m. Further, it can also identify transition of different bursts. Thus, it is called burst synchronization transition parameter.9/26/202445Dependence of the burst synchronization parameter m on $tau$ for different values of diffusion D9/26/202446Dependence of the burst synchronization parameter m on $tau$ for different values of the external stimulus I9/26/202447Resonance Resonance of a single HH neuron9/26/202448Resonance9/26/202449Spatial resonance of networks9/26/202450Spatial coherence resonanceModel Set-UpHodgkin and Huxley modelTransmission delaynoisethe Euler Maruyama method 9/26/202451Effect of delayAs the noise increases, pattern is destroyed. There is an intermediate noise, circular pattern can appear.9/26/202452Spatial structure function:Circular average:Signal-noise-ratio:Two dimensional Fourier Transform 9/26/202453Spatial structure function 9/26/202454Circular average and SNRMaximum indicating coherence9/26/202455For different delays, dependence of SNR on noise.9/26/202456Effect of small world connectionAs q is increased, Snr decreases.9/26/202457Open problems?Dynamical mechanism of synchronization bifurcation: synchronization-antiphase synchronization-synchronization as the delay/link probability is varied on small world/scale free networks.?Dynamical bifurcation mechanism of bursting firing should be explored as the delay is increased.?Theory exploration of singular phase locking.?Dynamics of more realistic neuronal networks9/26/2024581.Q.Y. Wang, Z.S. Duan, L. Huang, G.R. Chen, Q.S. Lu, Pattern formation and firing synchronization in networks of map neurons, New Journal of Physics, 9(383): 1-11, 2007.2.Q.Y. Wang, Z.S. Duan, Z.S. Feng, G.R. Chen and Q.S. Lu，Synchronization transition in Gap-Junction-Coupled leech neurons, Physica A, 2008, 387 4404-4410. 3.Q. Y. Wang, Matja Perc, Zhisheng Duan, Guanrong Chen, Delay-enhanced coherence resonance of spiral wave in noisy Hodgkin-Huxley neuronal networks, Phys. Lett.A, 372 (2008) 5681.4.Qingyun Wang, Zhisheng Duan, Matja Perc, Guanrong Chen, Synchronization transitions on small-world neuronal networks: effects of information transmission delay and rewiring probability, Europhysics Letters, 83 (2008) 500085.Qingyun Wang, Matjaz Perc, Zhi Sheng Duan, Guan Rong Chen, Synchronization transitions on scale-free neuronal networks due to finite information transmission delays, Physical Review E, 80, 026206,2009. 6.Qingyun Wang, Matjaz Perc, Zhisheng Duan, Guanrong Chen, Delay-induced multiple stochastic resonances on scale-free neuronal networks, Chaos, 19, 023112 2009 References9/26/202459oThis work was supported by the National Science Foundation of China (Funds No. 10702023, 10832006 and 10972001), China Doctoral Science Special Foundation (Fund No. 200801020).9/26/202460Thanks for your interesting!9/26/202461