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BME 6938 Neurodynamics. 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.

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Bme 6938 neurodynamics

BME 6938Neurodynamics

Instructor: Dr Sachin S Talathi

Isochrons define phase off limit cycle
Isochrons-Define phase off limit cycle

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

Weak coupling infinitesimal prc
Weak Coupling- Infinitesimal PRC

Linear Response Function or iPRC

Practical approach to calculating 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 equation

    Where with

    (XPPAUTO exploits this theorem to estimate iPRC)

Brain rhythms eeg correlate with behavioral states
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

Neural synchrony mechanism for generation of brain rhythms
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
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
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
Weakly coupled oscillators




Two weakly coupled oscillators
Two weakly coupled oscillators

represents deviation from the identical period for each oscillator

Analyze simple network weak coupled oscillators
Analyze Simple network-Weak Coupled Oscillators

Phase Locked Solution:

Stability Criteria:

Spike time response curves
Spike Time Response Curves

Coupling parameters

perturbation time

Intrinsic period

Analysis of the network using strcs
Analysis of the network using STRCs

Phase Locked Solution:

Stability Criterion: