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Bistability resulting from rebound firing

Overview

These examples examine coexistence of resting and spiking behavior, rebound firing, and hysteresis caused by slow gates. They contrast HH and Erisir models with RTM neurons carrying either h-current or M-current feedback.

Core ideas

Bistability means the same drive supports more than one attracting behavior; the initial condition selects rest or sustained firing. A slowly evolving gate shifts effective excitability and can produce rebound after inhibition. Forward and backward f--I sweeps expose hysteresis, while gate plots compare actual gates to their voltage-dependent steady-state values.

Essential model

The added h-current has the form

$$ I_h=g_hr(v_h-v),\qquad \dot r=\frac{r_\infty(v)-r}{\tau_r(v)}, $$

where $r$ is a slow activation gate, $g_h$ is maximal conductance, and $v_h$ is its reversal potential. The M-current analog is $I_M=g_Mw(v_K-v)$ with a slow potassium gate $w$.

Code examples

All sixteen examples now live in one notebook, chapter18.ipynb: simulate_hh_bistable/_gates/_limited_n compare resting and firing HH trajectories, plot HH gates and their targets, and isolate the potassium gate's role; simulate_erisir_bistable/_gates/_limited_h do the same for the Erisir model; simulate_h_current and simulate_modified_tau_r plot the h-gate steady state and (modified) time constant; simulate_rtm_voltage_trace_with_i_h simulates a rebound-firing RTM trace; simulate_rtm_f_i_curve_with_i_h computes h-current RTM forward/backward f-I branches; simulate_rtm_with_i_h_bistable/_gates/ simulate_rtm_with_i_h_limited_r do the h-current RTM bistability family; simulate_rtm_with_i_m_bistable/_gates/simulate_rtm_with_i_m_limited_w do the M-current RTM family.

What to look for

Start each family with its _BISTABLE trace, then use _GATES to see which slow variable differs from its instantaneous target. Compare h-current and M-current RTM families: the former supplies depolarizing rebound feedback, whereas the latter is a slow outward brake. The f--I curve identifies the hysteretic interval.

Suggested order

  1. Run H_CURRENT, PLOT_MODIFIED_TAU_R, and RTM_VOLTAGE_TRACE_WITH_I_H.
  2. Compare the three HH and three Erisir examples.
  3. Compare the h-current and M-current RTM bistability families, then the f--I curve.

Prerequisites and related chapters

Chapter 09 covers slow adaptation currents, Chapter 14 explains unstable cycles and Chapter 17 introduces forward/backward f--I scans. Chapter 19 uses slow-current feedback on an even longer time scale to generate bursting.

Running the examples

Open chapter18.ipynb in Jupyter, or via the Colab badge at the top of the notebook, and run all cells top to bottom. The simulate_rtm_f_i_curve_with_i_h cell is very slow (several minutes to tens of minutes) -- the notebook notes this inline.