1 Noise-Induced Bursting Synchronization in A Population of Coupled Neurons  Population Synchronization  Examples Flashing of Fireflies, Chirping of Crickets, Brain Rhythms, Heartbeats, Circadian Rhythms Synchronized Flashing of Fireflies

22 Circadian Rhythms  Biological Clock Ensemble of Neurons in the Suprachiasmatic Nuclei (SCN) Located within the Hypothalamus: Synchronization → Circadian Pacemaker Time of day (h) Temperature (ºc) Growth Hormone (ng/mL)

33 Neural Synchronization Correlated with Brain Functions  Perception of Smell Synchronization in a Subgroup of Neurons in the Olfactory Bulb which are responsible to the given smell (gamma wave: Hz)  Visual Perception and Binding Synchronization within a Subgroup of Neurons in the Visual Cortex with similar stimulus specificity (gamma wave) Feature Integration: Binding of the Integrated whole Image via Synchronization in Groups of Neurons (that detect different local features of a visual object)  Spatial Navigation (Hippocampus) Subset of Hippocampal Neurons: Encoding the Current Spatial Location via Synchronization (theta wave: 4-10Hz) (1) Normal Rhythms (Efficient for Sensory and Cognitive Processing) (2) Pathological Rhythms (Associated with Neural Diseases) Low-frequency and Extremely Synchronous Rhythmic Activity [Epileptic Seizures, Tremors for the Parkinson Disease] Visual Cortex Hippocampus Olfactory Bulb

4 Deterministic Firings of the Neuron Spikings of the Suprathreshold Neuron (Information: Encoded in the neural spikes) Burstings of the Suprathreshold Neuron Silent (resting) State Spiking State Subthreshold Neuron Suprathreshold Neuron Subthreshold NeuronSuprathreshold Neuron Silent State Bursting State Bursting: Alternation between a silent phase and a active (bursting) phase of repetitive spikes (i.e., a neuron repeatedly fires bursts of spikes). One importance of burstings: necessary to overcome the synaptic transmission failure. Bursting neurons: cortical intrinsically bursting neurons, thalamocortical relay neurons, thalmic reticular neurons, hippocampal pyramidal neurons, purkinje cells in the cerebellum

5 Noise-Induced Firings of the Neuron Noise-Induced Burstings of the Subthreshold Neuron in A Noisy Environment Noise-Induced Spikings of the Subthreshold Neuron in A Noisy Environment

6 Noise-Induced Coherence Noise: Usually Regarded as a Nuisance, Degrading the Performance of Dynamical Systems Constructive Role of Noise: Emergence of Noise-Induced Dynamical Order  Stochastic Resonance [L. Gammaitoni et al. Rev. Mod. Phys. 70, 223 (1998) Appearance of A Peak  Noise-Induced Bursting Coherence (Our Concern) Noise-Enhanced Detection of Weak Periodic Signal Coherence between Noise-Induced Neural Burstings Raster Plot of Neural Spikes: Spatiotemporal Plot of Neural Spikes

7 Neural Systems Neural Signal (Electric Spikes) Stimulus  Sensory Spikes (Neurons)  Brain  Motor Spikes (Neurons)  Response [Strength of a Stimulus > Threshold  Generation and Transmission of Spikes by Neurons] Neurons  ~ (~100 billion) neurons in our brain [cf. No. of stars in the Milky Way ~ 400 billion] (Typical Size of a Neuron ~ 30  m, Each neuron has 10 3 ~ 10 4 synaptic connection: Synaptic Coupling) Electrode signal (mV)  Sum of the input signals at the Axon Hillock Sum > Threshold  Generation of a Spike  Synaptic Coupling  Excitatory Synapse  Exciting the Postsynaptic Neuron  Inhibitory Synapse  Inhibiting the Generation of Spikes of the Postsynaptic Neuron

8 A Series of Five Papers: Published in J. Physiol. (1952) (First four papers: experimental articles Conductance-Based Physiological Model: Suggested in the fifth article) Nobel Prize (1963) Unveiling the Key Properties of the Ionic Conductances Underlying the Nerve Spikes  One of The Great Achievements of The 20th-Century Biophysics Hodgkin-Huxley Model for the Squid Giant Axon Brain 1st-level neuron 2nd-level neuron 3rd-level neuron Giant axon Stellate nerve Smaller axons Cross section 1mm Stellate nerve with giant axon (A) (C) (B) Postsynaptic (3rd level) Presynaptic (2nd level) Stellate ganglion Squid giant axon = 800  m diameter Mammalian axon = 2  m diameter

9 Generation of Action Potentials (Spikes) Cell Membrane: a leaky capacitor (lipid bilayer) penetrated by ion (conducting) channels.  Non-Gated Channels  Resting Potential (~ -65mV)  Voltage-Gated Channels  Spikes (Action Potential ~ 30mV) Action Potential (Spike) Activation potential Na + conductance E Na EKEK 0 50 – Membrane potential (mV) Open channels per  m 2 of membrane Conductance: Voltage-Dependent K + conductance Membrane Potential V m  V in - V out Neuron: Excited  Generation of Spikes In Out K+K+ Na + Cl  Activation gate Inactivation gate Activation gate Na + KK 1. Resting State 2. Activated State 3. Inactivated State Na + channel K + channel Activation gate Na + K+K+ Inactivation gate Slow – – + + Out In (input signal > threshold)

10 Synaptic Coupling Synaptic Transmission Synaptic Coupling Type Arrival of Spikings at the Axon Terminal  Release of Chemical Transmitter at the Axon Terminal of the Presynaptic Neuron Open of the Receptor Channel of the Postsynaptic Neuron through the Binding of the Chemical Transmitter Receptor channel Postsynaptic neuron Presynaptic neuron … … Chemical Transmitter Na + K+K+  Inhibitory Synapse: Inhibiting the Generation of Spikes of the Postsynaptic Neuron (e.g., GABA Transmitter + GABA A /GABA B Receptors)  Excitatory Synapse: Exciting the Postsynaptic Neuron (e.g., Glutamate Transmitter + AMPA/NMDA Receptors)

11 Izhikevich Neuron Model Izhikevich Model [Biologically Plausible and Computationally Efficient] [E. M. Izhikevich, IEEE Trans. Neural Networks 14, 1569 (2003)] v: membrane potential u: recovery variable providing a negative feedback to v Parameters: a = 0.02, b = 0.2, c = - 65, d = - 8 I ext = I DC (Constant Bias) Average Firing Frequency with the auxiliary after-spike resetting :

12 Firings of the Izhikevich Neuron Firings of the Suprathreshold Neuron (Corresponding to Relaxation Oscillations) Noise-Induced Firings of the Subthreshold Neuron Silent State Spiking State Noisy Environment: I ext = I DC +D ,  : Gaussian white noise with =0 and =  (t  t ’ ).

13 Population of Subthreshold Izhikevich Neurons Global Synaptic Coupling (Each Neuron: Coupled to All the Other Neurons with Equal Strength) I DC =3.6  Neurons: Set in the subthreshold regime I syn,i : Synaptic current flowing into the ith neuron N: Total No. of Neurons, J/(N-1): maximal conductance per each synapse, s: synaptic gating variable representing fraction of open channels, s(t)  [0,1], V syn : synaptic reversal potential  : synaptic opening rate (=inverse of the synaptic rise time)  : synaptic closing rate (=inverse of the synaptic decay time) s  (v): Normalized concentration of neurotransmitters modeled by the sigmoidal (Boltzmann) function C + [ T ] O   Excitatory Synapse with AMPA Receptors Neurotransmitter: Glutamate Receptor: AMPA   =10 ms -1 (  r =0.1 ms),  =0.5 ms -1 (  d =2 ms), V syn =0 mV

14 Noise–Induced Burstings Bursting Activity Alternating between The Active Phase (repetitive spikings) and The Quiescent Phase v 1 : fast membrane potential variable. u 1 : slow recovery variable providing a negative feedback to v 1 (u 1 : min → active phase, u 1 : max → quiescent phase J = 1.5 and D = 0.5 Occurrence of Bursting Activity on A Hedgehoglike Limit Cycle (Spine : Active phase, Body: Quiescent phase)

15 Noise-Induced Bursting Synchronization  Characterization of noise-Induced Burst Synchronization in a Population of 10 3 Globally Coupled Neurons for J=1.5 Description of Emergence of Collective Bursting Synchronization in Terms of Population-Averaged Membrane Global Potential V G and The Global Recovery Variable U G :  Visualization of Noise-Induced Burst Synchronization (Collective Coherence between Noise-Induced Burstings) in the Raster Plot of spikes D Incoherence Noise-Induced Bursting Sync (Onset of Bursting Sync. Because of the Constructive Role of Noise to Stimulate Coherence between Noise-Induced Bursting) (Disappearance of Bursting Sync Due to The Destructive Role of Noise to Spoil the Bursting Sync) [S.-Y. Kim, Y. Kim, D.-G. Hong, J. Kim, and W. Lim, J. Korean Phys. Soc. 60, 1441 (2012)]

16 Transition from Burst/Spike Sync. to Burst Sync. Burst/Spike Sync Burst Sync D  Burst/Spike Synchronization (1) D = 0.2 Appearance of Clear Burst Bands at Regular Time Intervals (Burst Sync) Each Burst Band : Composed of Stripes of Spikes (Spike Sync) → V G : Bursting Activity (Fast Spikes on A Slow Wave) (2) D = 0.5  Burst Synchronization (3) D = 12 (4) D = 17 Smearing of Stripes in Each Burst Band → Amplitude of Spikes in V G : Decreased Loss of Spike Sync in Each Burst Band →V G : Slow Wave Without Spikes Burst Band : Smearing with further Increase in D, Overlapping of Burst Bands → Incoherent state

17 Summary Emergence of Noise-Induced Bursting Synchronization Appearance of Collective Coherence between Noise-Induced Burstings via the Competition of the Constructive and the Destructive Roles of Noise Burst/Spike Sync Burst Sync D Incoherence Noise-Induced Burst Sync  Burst/Spike Synchronization  Burst Synchronization