1. Sakai, Yamada & Yoshizawa 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. Sakai, Yamada & Yoshizawa Neuron: excitable system ～ bifurcation type ～ Behavior for constant current injection silent → periodic firing LIF, HH+A-current (saddle node bifurcation) frequency μ Continuous HH (Hopf bifurcation) frequency μ Discontinuous

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

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

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

6. Sakai, Yamada & Yoshizawa Spike Sequences of Cortical regular spiking neurons | | || | | | | | Highly Variable Intervals Fluctuating input???

7. Sakai, Yamada & Yoshizawa Naïve expectation Higher Variability in ISI sequence | | || | | | | | Stochastic factor increase Stochastic factor increase Higher balance of fluctuation Stochastic factor increase Stochastic factor increase ??? input output

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

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

10. Sakai, Yamada & Yoshizawa Difference HH v.s. LIF in CV-σ

11. Sakai, Yamada & Yoshizawa Difference: HH v.s. LIF LIF can never reproduce “monotone decreasing” at any parameter range! at any refractory! Output Variability for Input Fluctuation LIF: monotone increasing HH: monotone decreasing

12. Sakai, Yamada & Yoshizawa HH + A-current (CWM model) monotone increasing

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

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

15. Sakai, Yamada & Yoshizawa Type of Stable Fixed point ～ Typical behavior before bifurcation near saddle-node bifurcation bifurcation near Hopf bifurcation

16. Sakai, Yamada & Yoshizawa Essences of the mechanism 1. Discontinuous jump of firing frequency 2. Second firing for a single perturbation 3. Refractory

17. Sakai, Yamada & Yoshizawa Mechanism of decreasing Variability Near before bifurcation Poisson Bursting Pattern || || ||| || Far before bifurcation Poisson + Ref. Pattern | | | |

18. Sakai, Yamada & Yoshizawa Suggestion Higher Variability in ISI sequence | | || | | | | | Higher balance of fluctuation input output Does Not Always mean

19. Sakai, Yamada & Yoshizawa Difference between Hopf & Saddle-node Throughout concerned parameter range, mean input μ lies in before the bifurcation point Before Saddle-node bifurcation non-firing spiral, non-spiral non-firing spiral, or non-spiral stable fix point Before Hopf bifurcation firing spiral firing spiral stable fix point

20. Sakai, Yamada & Yoshizawa Firing Spiral Stable Fix point ～ Typical before Hopf bifurcation

21. Sakai, Yamada & Yoshizawa CV-σ ( μ: const ) Jump