BME 6938Neurodynamics

Instructor: Dr Sachin S Talathi

Phase of limit cycle

Isochrons-Define phase off limit cycle

Phase of a non periodic point is taken to be the phase of its periodic proxy

Phase Response Curve

Weak Coupling- Infinitesimal PRC

Linear Response Function or iPRC

Practical Approach to Calculating iPRC Malkin’s Theorem:Let the system have exponentially

stable limit cycle with period T and receive infinitesimal periodic perturbation

Then its phase is described by equationWhere with

(XPPAUTO exploits this theorem to estimate iPRC)

Brain rhythms (EEG) correlate with behavioral states•Delta (0.5-4 Hz):Dominant rhythm in infants and stage 3 and 4 of sleep

•Theta (4-8 Hz):Normal activity in young children and represents drowsiness in adults

•Alpha (8-12 Hz):It is observed in relaxed state

•Beta (12-30 Hz): Observed in an anxious state

•Gamma (>30 Hz): Observed in attention state and is thought to be the learning rhythm

Deep Sleep

Drowsy

Relaxed

Excited

Neural synchrony: Mechanism for generation of brain rhythms

Synchronous activity is large-detectable at the electrodes on the scalp (source of EEG)

Neural Synchrony and the Binding Problem No central location in the brain where all information

related to a task is centralized

How are the parallel computations in spatially segregated regions in the brain coordinated?

How are signals selected and routed from sensory structures to executive structures without confounding?

How information about relatedness of content is encoded?

Related to the problem of consciousness

Potential Answer: Neural synchrony

How does synchrony arise?Two key mechanisms.

Related to the intrinsic properties of neurons in terms their preference for input frequencies (resonance)

Related to the pattern of connectivity between neurons and the dynamic properties of intervening synapses (network and network interactions)

Note: These are not mutually exclusive explanations

Weakly coupled oscillators

Substitute

where

Note

Two weakly coupled oscillators

represents deviation from the identical period for each oscillator

Analyze Simple network-Weak Coupled Oscillators

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dφA

dt=ω A + H Δφ( )

dφB

dt=ωB

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dΔφdt

=Δω −H Δφ( )Stability Criteria:

Phase Locked Solution:

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H Δφ*( )=Δω

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dH Δφ( )dΔφ

Δφ*

> 0

Results from Weak Coupling Theory Analysis

Spike Time Response Curves

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Φ j δt,T0,E I ,gs,τ R ,τ D( ) =Tj − T0

T0

perturbation time

Intrinsic period

Coupling parameters

Analysis of the network using STRCs

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δ n +1 = δ n + TB − TA 1+ Φ1 δ n( ) + Φ 2 δ n( )( )

Phase Locked Solution:

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1+ Φ1 δ *( ) + Φ 2 δ *( ) =TB

TA

Stability Criterion:

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0 <dΦ ∞ x( )

dxx =δ *

< 2

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Φ∞ x( ) = Φ1 x( ) + Φ 2 x( )

Results from analysis using STRCs