# hh_cond_beta_gap_traub – Hodgkin-Huxley neuron with gap junction support and beta function synaptic conductances¶

## Description¶

hh_cond_beta_gap_traub is an implementation of a modified Hodgkin-Huxley model that also supports gap junctions.

This model was specifically developed for a major review of simulators 1, based on a model of hippocampal pyramidal cells by Traub and Miles 2. The key differences between the current model and the model in 2 are:

This model is a point neuron, not a compartmental model.

This model includes only I_Na and I_K, with simpler I_K dynamics than in 2, so it has only three instead of eight gating variables; in particular, all Ca dynamics have been removed.

Incoming spikes induce an instantaneous conductance change followed by exponential decay instead of activation over time.

This model is primarily provided as reference implementation for hh_coba example of the Brette et al (2007) review. Default parameter values are chosen to match those used with NEST 1.9.10 when preparing data for 1. Code for all simulators covered is available from ModelDB 3.

Note: In this model, a spike is emitted if \(V_m \geq V_T + 30\) mV and \(V_m\) has fallen during the current time step.

To avoid that this leads to multiple spikes during the falling flank of a spike, it is essential to chose a sufficiently long refractory period. Traub and Miles used \(t_{ref} = 3\) ms (2, p 118), while we used \(t_{ref} = 2\) ms in 2.

Postsynaptic currents Incoming spike events induce a postsynaptic change of conductance modelled by a beta function as outlined in 4 5. The beta function is normalized such that an event of weight 1.0 results in a peak current of 1 nS at \(t = \tau_{rise,xx}\) where xx is ex or in.

Spike Detection Spike detection is done by a combined threshold-and-local-maximum search: if there is a local maximum above a certain threshold of the membrane potential, it is considered a spike.

Gap Junctions Gap Junctions are implemented by a gap current of the form \(g_{ij}( V_i - V_j)\).

## Parameters¶

The following parameters can be set in the status dictionary.

V_m |
mV |
Membrane potential |

V_T |
mV |
Voltage offset that controls dynamics. For default parameters, V_T = -63mV results in a threshold around -50mV |

E_L |
mV |
Leak reversal potential |

C_m |
pF |
Capacity of the membrane |

g_L |
nS |
Leak conductance |

tau_rise_ex |
ms |
Excitatory synaptic beta function rise time |

tau_decay_ex |
ms |
Excitatory synaptic beta function decay time |

tau_rise_in |
ms |
Inhibitory synaptic beta function rise time |

tau_decay_in |
ms |
Inhibitory synaptic beta function decay time |

t_ref |
ms |
Duration of refractory period (see Note) |

E_ex |
mV |
Excitatory synaptic reversal potential |

E_in |
mV |
Inhibitory synaptic reversal potential |

E_Na |
mV |
Sodium reversal potential |

g_Na |
nS |
Sodium peak conductance |

E_K |
mV |
Potassium reversal potential |

g_K |
nS |
Potassium peak conductance |

I_e |
pA |
External input current |

## References¶

- 1(1,2)
Brette R et al (2007). Simulation of networks of spiking neurons: A review of tools and strategies. Journal of Computational Neuroscience 23:349-98. DOI: https://doi.org/10.1007/s10827-007-0038-6

- 2(1,2,3,4,5)
Traub RD and Miles R (1991). Neuronal Networks of the Hippocampus. Cambridge University Press, Cambridge UK.

- 3
- 4
Rotter S and Diesmann M (1999). Exact digital simulation of time-invariant linear systems with applications to neuronal modeling. Biological Cybernetics 81:381 DOI: https://doi.org/10.1007/s004220050570

- 5
Roth A and van Rossum M (2010). Chapter 6: Modeling synapses. in De Schutter, Computational Modeling Methods for Neuroscientists, MIT Press.

## Sends¶

SpikeEvent

## Receives¶

SpikeEvent, CurrentEvent, DataLoggingRequest