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Strange responses of the Hodgkin-Huxley model to highly fluctuating inputs. Yutaka Sakai ( Saitama Univ., Japan ) Masahiro Yamada ( Univ. Tokyo, Japan ) Shuji Yoshizawa ( Saitama Univ., Japan ). class II. HH (Hopf bifurcation). frequency. frequency.
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Strange responses of the Hodgkin-Huxley model to highly fluctuating inputs Yutaka Sakai (Saitama Univ., Japan) Masahiro Yamada (Univ. Tokyo, Japan) Shuji Yoshizawa (Saitama Univ., Japan)
class II HH (Hopf bifurcation) frequency frequency Discontinuous μ μ LIF, HH+A-current (saddle node bifurcation) class I Continuous Neuron: excitable system ~ bifurcation type ~ Behavior for constant current injection silent → periodic firing
Leaky integrate-and-fire(LIF) Model Hodgkin-Huxley(HH) Model Neuron Model (Hodgkin & Huxley 1952: squid axon)
: transient, inactivating HH + A-current (CWM Model) Connor-Walter-McKown(CWM) Model (Connor, Walter & McKown 1977: crab axon )
Saddle-node bifurcation Hopf bifurcation Effect of A-current (Inactivating, ) 2D phase space W V
??? | | || | | | | | Fluctuating input Spike Sequences of Cortical regular spiking neurons Highly Variable Intervals
Higher balance of fluctuation Stochastic factor increase Higher Variability in ISI sequence | | || | | | | | • Stochastic factor increase Naïve expectation input ??? output
Spike sequence | | || | | | | | T Response to fluctuate input Neuron Model Inter-Spike Interval (ISI) Statistics Mean Interval Coefficient of Variation T : Inter-Spike Interval (ISI)
Effective Strength of input : const. adjust const. Relationship Relationship:“Input fluctuation”– “Output variability” fluctuation LIF or HH variability CV
Difference: HH v.s. LIF Output Variability for Input Fluctuation HH: monotone decreasing LIF: monotone increasing LIF can never reproduce “monotone decreasing” at any parameter range! at any refractory!
Summary of Results Output Variability for Input Fluctuation • LIF: monotone increasing • HH: monotone decreasing (Hopf bifurcation) + A-current (Saddle-node bifurcation) • CWM: monotone increasing
Suggestion of Result The strange response of HH : “monotone decreasing variability” seems to originate in Property of Hopf bifurcation . . . Why?
Type of Stable Fixed point ~ Typical behavior before bifurcation class I near saddle-node bifurcation class II near Hopf bifurcation
Essences of the mechanism 1. Discontinuous jump of firing frequency 2. Second firing for a single perturbation 3. Refractory
Near before bifurcation Poisson Bursting Pattern || || ||| || Far before bifurcation Poisson + Ref. Pattern | | | | | | | | Mechanism of decreasing Variability
Higher balance of fluctuation Higher Variability in ISI sequence | | || | | | | | Suggestion input Does Not Always mean output
Difference between Hopf & Saddle-node Throughout concerned parameter range, mean input μ lies inbefore the bifurcation point Before Hopf bifurcation firing spiralstable fix point Before Saddle-node bifurcation non-firing spiral, ornon-spiralstable fix point
Firing Spiral Stable Fix point ~ Typical before Hopf bifurcation
CV-σ ( μ: const ) Jump