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As a passion project I'm recreating a neuronal model from XJ Wang's lab at NYU. The paper is Wei, W., & Wang, X. J. (2016). Inhibitory control in the cortico-basal ganglia-thalamocortical loop: complex regulation and interplay with memory and decision processes. Neuron, 92(5), 1093-1105.

The main problem I'm having is interpreting the equation for calculating the differential of the neurons membrane voltage. They have included a bursting neuronal model for cells in the basal ganglia and subthalamic nucleus. The differential equation for membrane voltage in these regions incorporates a hyperpolarization rebound which results in bursts and tonic spiking. The equation is on page 2 of a prior paper which uses basically the exact same model. I have linked to the paper below and I have provided an image link to the exact passage as well. http://www.cns.nyu.edu/wanglab/publications/pdf/wei.jns2015.pdf

This is the equation I'm having trouble reading, don't worry about Isyn its the input current from the synapses

The equation is taken from this paper: https://www.physiology.org/doi/pdf/10.1152/jn.2000.83.1.588

Obviously the equation will need to be discritized so I can run it with numpy but I ill ignore that for now as it will be relatively easy to do so. The middle term with all the H's is whats giving me trouble. As I understand it I should be running code which dos the following:

gt * h * H(V-Vh) * (V-Vt)

Where H(V-Vh) is the heavyside step function, V is the membrane voltage at the prior timestep Vh = -60mV and Vt = 120mV. gt is a conductance efficacy constant in nanoSiemens. I think the correct way to interpret this for python is...

gt * h * heavyside(-60, 0.5)*(V-120)

But I'm not 100% sure I'm reading the notation correctly. Could someone please confirm I've read it as it is intended?

Secondly h is the deactivation term which gives rise to bursting as described in the final paragraph on page 2 of Smith et al., 2000 (the second pdf I've linked to). I understand the differential equations that govern the evolution of h well enough but what is the value of h? In Smith et al. 2000 the authors say that h relaxes to zero with a time constant of 20ms and it relaxes to unity with a time constant of 100ms. What value is h relaxing from and what does it mean to relax to unity?

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Angus Campbell
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  • Update: I now understand my second question. The differential for deinactivation above bursting threshold is (1-h)/tau where tau=100ms. This means that at rest h=h+dt*0.01 which makes the term fairly negligable compared to the other currents. I can initialize h at 0 and the model should be fine. – Angus Campbell Mar 12 '18 at 18:17

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For you x1 (of the numpy.heaviside) is = V-Vh; you are comparing that difference to zero. You might try writing your own version of the Heaviside function to deepen understanding, and then move back to the numpy version if you need it for speed or compatibility. The pseudo-code wordy version would be something like,

if (V<Vh): return(0); else: return(1);

You could probably just write (V>=Vh) in your code as Python will treat the boolean as 1 if true and 0 if false.

This ignores the possibility of V==Vh in the complete version of Heaviside, but for most practical work with real values (even discretized in a computer) that is unlikely to be a case to worth concerning yourself with, but you could easily add it in.

brittAnderson
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