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Neuroscience: poisson and gauss in neuron firing rate model 
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#1
Feb1113, 02:33 PM

P: 503

Hello!
I was reading a journal article on modeling the interaction between different neural networks and I am confused about the follwoing method (cited below). It is describing the spike rate output of a neuron based on oscillating firing rates of excitatory (E) and inhibitory (I) inputs: "Consider two local circuits, both projecting to a third circuit ... each comprised of E and I cells, with at least a projection from the local I cells to the E cells. When an input network is synchronized it produces periodic E cell activity at a specific global phase set by its local I cells. These two sources of E volleys together with the local inhibition drive the E cells in the receiving circuit. Here we are interested in modeling the impact of E and I streams that are out of phase. We studied the effect of synchronized E and I inputs on a model neuron with Hodgkin–Huxleytype channels.... Periodic and synchronous activity was modeled as a Poisson process with a timevarying firing rate comprised of a periodic sequence of Gaussian peaks. Each Gaussian peak generated a socalled volley: a set of input spike times tightly centered on the location of the peak." and "The number of incoming I and E spikes varied from cycle to cycle because the E and I inputs were generated as Poisson processes with a spike density comprised of a periodic sequence of Gaussian peaks." Is it describing a nonhomogeneous Poisson process which, due to the nature of the timevarying probability of a spike occurrence, produces a firing rate for the inputs that looks like a series of (approximate) 'Gaussianshaped' curves? Or is there something I am missing in the "Gaussian peak" part? I initially thought that the peaks were generated through a Poisson process, and then 'something else happened', involving generating "volleys" via a different probability distribution. But the peaks are evenly spaced at a constant 25 ms period, and the actual number of spikes, as described above, varies due to it being a Poisson process. So, I figured that the actual spike occurrence at any point in time is given by the timevarying Poisson distribution, and the nature of the function, with timevarying λ(t), produces a series of spikes that looks like, as mentioned before, a series of 'Gaussianshaped curves'. Otherwise, I don't see where the Poisson process part comes in. Paper: Mechanisms for Phase Shifting in Cortical Networks and their Role in Communication through Coherence by Paul H. Tiesinga and Terrence J. Sejnowski. Any help appreciated. 


#2
Feb1113, 02:56 PM

HW Helper
P: 1,391

It sounds to me like the firing rate has a Gaussian profile, i.e., ##\lambda(t) \propto \exp((tt_0)^2/\sigma^2)##. The probability of a spike is highest in a timewindow containing the peak of the firing rate, so you find that most spikes are produced when the Gaussian is peaked. That doesn't mean that the spikes themselves appear to form a Gaussianshaped curve. Presumably all of the spikes have the same height. However, if you plotted the density of the spikes (how close together they are), then the density profile would look roughly Gaussian.



#3
Feb1513, 02:58 AM

P: 503

Hi Mute, thanks for the response.
"Presumably all of the spikes have the same height." That's my bad, I meant the firing rate. I need to get my bearings. The probability of a spike, the firing rate and the spike density are all related: the higher the probability of firing the higher the firing rate (generally), the higher the firing rate the greater the spike density. In the case of the timedependent firing rate, r(t), the spike density provides a means of computing r(t). The λ(t) comes from the Poisson distribution for the probability of firing, right? In this case, the probability is timedependent; and because of the nature of the changes in probability of firing over time, the firing rate appears as a series of 'Gaussianshaped curves'. Also, since the firing rate follows a Gaussianshaped curve, the spike density is similarly Gaussianshaped. I think this is what you were saying in your previous post. As an addition to this, the paper also states that a constant depolarising current is applied, I was wondering if this is common in modelling neurons,; it means that a constant depolarising current, a periodic inhibitory current, and a periodic excitatory current are all applied. I don't understand the point of applying the constant depolarising current. Thanks for the help so far, any further help appreciated. 


#4
Feb1513, 06:31 AM

Sci Advisor
P: 8,367

Neuroscience: poisson and gauss in neuron firing rate model



#5
Feb2113, 04:09 PM

P: 503

Thanks for the help so far. 


#6
Feb2413, 10:16 AM

Sci Advisor
P: 8,367

Take Figure 2 for example. The neuron receives excitation and inhibition governed by periodic processes (consisting of a bunch of periodic Gaussianlike bumps). In addition to these two inputs, it receives a constant current which is termed the "driving current". In general, excitation increases the probability of spiking, so it would not be wrong to say that the periodic Gaussianlike excitation is also a driving current (or at least a driving input), but in Fig 2 they reserve that term for the constant current. 


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