Noise-Induced Bursting Synchronization in A Population of Coupled Neurons

1. Noise-InducedBursting 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

2. Circadian Rhythms Growth Hormone (ng/mL) Temperature (ºc) Time of day (h) Time of day (h) • Biological Clock Ensemble of Neurons in the Suprachiasmatic Nuclei (SCN) Located within the Hypothalamus: Synchronization → Circadian Pacemaker 2

3. Neural Synchronization Correlated with Brain Functions (1) NormalRhythms (Efficient for Sensory and Cognitive Processing)  Perception of Smell Visual Cortex Synchronization in a Subgroup of Neurons in the Olfactory Bulb which are responsible to the given smell (gamma wave: 30-80 Hz) Olfactory Bulb  Visual Perception and Binding Hippocampus 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) (2) Pathological Rhythms (Associated with Neural Diseases) Low-frequency and Extremely Synchronous Rhythmic Activity [Epileptic Seizures, Tremors for the Parkinson Disease] 3

4. Deterministic Firings of the Neuron •Spikings of the Suprathreshold Neuron (Information: Encoded in the neural spikes) Silent (resting) State Spiking State Suprathreshold Neuron Subthreshold Neuron • Burstings of the Suprathreshold Neuron 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 Silent State Bursting State Subthreshold Neuron Suprathreshold Neuron

5. Noise-Induced Firings of the Neuron •Noise-Induced Spikings of the Subthreshold Neuron in A Noisy Environment •Noise-Induced Burstings 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) Noise-Enhanced Detection of Weak Periodic Signal Appearance of A Peak  Noise-Induced Bursting Coherence (Our Concern) Raster Plot of Neural Spikes: Spatiotemporal Plot of Neural Spikes Coherence between Noise-Induced Neural Burstings 6

7. Neural Systems Electrode signal (mV) • 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  ~ 1011 (~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 103 ~ 104 synaptic connection: Synaptic Coupling)  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. Hodgkin-Huxley Model for the Squid Giant Axon Giant axon Brain Presynaptic (2nd level) Stellate nerve 1st-level neuron Smaller axons Stellate ganglion 2nd-level neuron 3rd-level neuron Postsynaptic (3rd level) Cross section Stellate nerve with giant axon 1mm 1mm (A) (B) (C) Squid giant axon = 800m diameter Mammalian axon = 2 m diameter • 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

9. Generation of Action Potentials (Spikes) Membrane Potential Vm  Vin - Vout Conductance: Voltage-Dependent ENa 50 Activation potential Open channels per m2 of membrane Membrane potential (mV) 0 Na+ conductance 40 2. Activated State 1. Resting State K+ conductance 20 –50 Na+ Na+ channel K+ channel 0 – – – – – – Out + EK + + + + Neuron: Excited  Generation of Spikes + + + + – – – In + + – – Inactivation gate Slow Activation gate 3. Inactivated State – – – – – – + + + + + + K+ Slow • Cell Membrane: a leaky capacitor (lipid bilayer) penetrated by ion (conducting) channels. Cl- Na+ • Non-Gated Channels  Resting Potential (~ -65mV) • Voltage-Gated Channels  Spikes (Action Potential ~ 30mV) Na+ Cl- Na+ Out Activation gate In K+ Activation gate Inactivation gate K+ (input signal > threshold) • Action Potential (Spike)

10. Synaptic Coupling … … • Synaptic Transmission Open of the Receptor Channel of the Postsynaptic Neuron through the Binding of the Chemical Transmitter • Arrival of Spikings at the Axon Terminal • Release of Chemical Transmitter at the Axon Terminal of the Presynaptic Neuron Presynaptic neuron Chemical Transmitter Na+ Receptor channel Postsynaptic neuron K+ • Synaptic Coupling Type • Excitatory Synapse: Exciting the Postsynaptic Neuron (e.g., Glutamate Transmitter + AMPA/NMDA Receptors)  Inhibitory Synapse: Inhibiting the Generation of Spikes of the Postsynaptic Neuron (e.g., GABA Transmitter + GABAA /GABAB 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 variableproviding a negative feedback to v Parameters: a = 0.02, b = 0.2, c = -65, d = -8 with the auxiliary after-spike resetting : Iext = IDC (Constant Bias) •Average Firing Frequency

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

13. Population of Subthreshold Izhikevich Neurons • Global Synaptic Coupling (Each Neuron: Coupled to All the Other Neurons with Equal Strength) IDC=3.6  Neurons: Set in the subthreshold regime Isyn,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], Vsyn: 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), Vsyn=0 mV

14. Noise–Induced Burstings • Bursting Activity Alternating between The Active Phase (repetitive spikings) and The Quiescent Phase v1 : fast membrane potential variable. u1 : slow recovery variable providing a negative feedback to v1 (u1 : min → active phase, u1 : 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 [S.-Y. Kim, Y. Kim, D.-G. Hong, J. Kim, and W. Lim, J. Korean Phys. Soc. 60, 1441 (2012)] • Characterization of noise-Induced Burst Synchronization in a Population of 103 Globally Coupled Neurons for J=1.5 • Description of Emergence of Collective Bursting Synchronization in Terms of Population-Averaged Membrane Global • Potential VG and The Global Recovery Variable UG : • Visualization of Noise-Induced Burst Synchronization (Collective Coherence between Noise-Induced Burstings) in the Raster Plot of spikes Incoherence Noise-Induced Bursting Sync Incoherence D (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)

16. Transition from Burst/Spike Sync. to Burst Sync. Burst Sync Burst/Spike 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) • → VG : Bursting Activity (Fast Spikes on A Slow Wave) (2) D = 0.5 Smearing of Stripes in Each Burst Band → Amplitude of Spikes in VG : Decreased • Burst Synchronization (3) D = 12 Loss of Spike Sync in Each Burst Band →VG : Slow Wave Without Spikes (4) D = 17 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 Noise-Induced Burst Sync Incoherence Burst Sync Incoherence Burst/Spike Sync D • Burst/Spike Synchronization • Burst Synchronization