Noise induced bursting synchronization in a population of coupled neurons
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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. Circadian Rhythms. Growth

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Noise-Induced Bursting Synchronization in A Population of Coupled Neurons

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Noise-InducedBursting Synchronization in A Population of Coupled Neurons

  • Population Synchronization


Flashing of Fireflies, Chirping of Crickets, Brain Rhythms, Heartbeats, Circadian Rhythms

Synchronized Flashing of Fireflies

Circadian Rhythms


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


Neural Synchronization Correlated with Brain Functions

(1) NormalRhythms (Efficient for Sensory and Cognitive Processing)

 Perception of Smell



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


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]


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

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

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


Neural Systems


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


 Inhibitory Synapse

 Inhibiting the Generation

of Spikes of the

Postsynaptic Neuron

Hodgkin-Huxley Model for the Squid Giant Axon

Giant axon



(2nd level)

Stellate nerve

1st-level neuron

Smaller axons



2nd-level neuron

3rd-level neuron


(3rd level)

Cross section


nerve with

giant axon






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

Generation of Action Potentials (Spikes)

Membrane Potential Vm  Vin - Vout

Conductance: Voltage-Dependent



Activation potential

Open channels per

m2 of membrane

Membrane potential (mV)


Na+ conductance


2. Activated State

1. Resting State

K+ conductance




Na+ channel

K+ channel


– –

– –

– –




+ +

+ +

Neuron: Excited  Generation of Spikes

+ +

+ +

– –


+ +

– –

Inactivation gate




3. Inactivated State

– –

– –

– –

+ +

+ +

+ +



• 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)





Activation gate








(input signal > threshold)

• Action Potential (Spike)

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


Receptor channel

Postsynaptic neuron


• 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)

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


a = 0.02, b = 0.2, c = -65, d = -8

with the auxiliary after-spike resetting :

Iext = IDC (Constant Bias)

•Average Firing Frequency

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’).

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

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)

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


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)

Transition from Burst/Spike Sync. to Burst Sync.

Burst Sync

Burst/Spike Sync


  • 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


• 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


Burst Sync


Burst/Spike Sync


  • Burst/Spike Synchronization

  • Burst Synchronization

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