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
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
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
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
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 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 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