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Strange responses of the Hodgkin-Huxley model to highly fluctuating inputs

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

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  1. 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)

  2. 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

  3. Leaky integrate-and-fire(LIF) Model Hodgkin-Huxley(HH) Model Neuron Model (Hodgkin & Huxley 1952: squid axon)

  4. ： transient, inactivating HH + A-current (CWM Model) Connor-Walter-McKown(CWM) Model (Connor, Walter & McKown 1977: crab axon )

  5. Saddle-node bifurcation Hopf bifurcation Effect of A-current (Inactivating, ) 2D phase space W V

  6. ??? | | || | | | | | Fluctuating input Spike Sequences of Cortical regular spiking neurons Highly Variable Intervals

  7. Higher balance of fluctuation Stochastic factor increase Higher Variability in ISI sequence | | || | | | | | • Stochastic factor increase Naïve expectation input ??? output

  8. Spike sequence | | || | | | | | T Response to fluctuate input Neuron Model Inter-Spike Interval (ISI) Statistics Mean Interval Coefficient of Variation T : Inter-Spike Interval (ISI)

  9. Effective Strength of input : const. adjust const. Relationship Relationship:“Input fluctuation”– “Output variability” fluctuation LIF or HH variability CV

  10. Difference HH v.s. LIF in CV-σ

  11. 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!

  12. HH + A-current (CWM model)monotone increasing

  13. Summary of Results Output Variability for Input Fluctuation • LIF: monotone increasing • HH: monotone decreasing (Hopf bifurcation) + A-current (Saddle-node bifurcation) • CWM: monotone increasing

  14. Suggestion of Result The strange response of HH : “monotone decreasing variability” seems to originate in Property of Hopf bifurcation . . . Why?

  15. Type of Stable Fixed point ～ Typical behavior before bifurcation class I near saddle-node bifurcation class II near Hopf bifurcation

  16. Essences of the mechanism 1. Discontinuous jump of firing frequency 2. Second firing for a single perturbation 3. Refractory

  17. Near before bifurcation Poisson Bursting Pattern || || ||| || Far before bifurcation Poisson + Ref. Pattern | | | | | | | | Mechanism of decreasing Variability

  18. Higher balance of fluctuation Higher Variability in ISI sequence | | || | | | | | Suggestion input Does Not Always mean output

  19. 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

  20. Firing Spiral Stable Fix point ～ Typical before Hopf bifurcation

  21. CV-σ ( μ: const ) Jump

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