# 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

http://modeldb.yale.edu/83319

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.

SpikeEvent