Neuroscience: poisson and gauss in neuron firing rate model

[nobahar]

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–Huxley-type channels... Periodic and synchronous activity was modeled as a Poisson process with a time-varying firing rate comprised of a periodic sequence of Gaussian peaks. Each Gaussian peak generated a so-called 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 non-homogeneous Poisson process which, due to the nature of the time-varying probability of a spike occurrence, produces a firing rate for the inputs that looks like a series of (approximate) 'Gaussian-shaped' 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 time-varying Poisson distribution, and the nature of the function, with time-varying λ(t), produces a series of spikes that looks like, as mentioned before, a series of 'Gaussian-shaped 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.

 

[Mute]

It sounds to me like the firing rate has a Gaussian profile, i.e., $\lambda(t) \propto \exp(-(t-t_0)^2/\sigma^2)$. The probability of a spike is highest in a time-window 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 Gaussian-shaped 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.

 

[nobahar]

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 time-dependent 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 time-dependent; and because of the nature of the changes in probability of firing over time, the firing rate appears as a series of 'Gaussian-shaped curves'. Also, since the firing rate follows a Gaussian-shaped curve, the spike density is similarly Gaussian-shaped.

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.

 

[atyy]

Is it describing a non-homogeneous Poisson process which, due to the nature of the time-varying probability of a spike occurrence, produces a firing rate for the inputs that looks like a series of (approximate) 'Gaussian-shaped' curves?

That's my understanding too.

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.

I think they envisage that the constant current could be produced by a neuromodulator in the extracellular fluid. Eg. in the paragraph "In summary, the global phase of a local circuit can be modulated by pulses, such as those mediated by a synchronized volley of synaptic inputs or optogenetic stimulation (Cardin et al., 2009; Sohal et al., 2009), whereas changing the internal phase between local E and I required a constant depolarization to be maintained for as long as the internal phase needs to have the altered value. Such slower time scale modulations could be mediated by neuromodulators (Buia and Tiesinga, 2006)."

 

[nobahar]

That's my understanding too.

That's reassuring! I'll take it to mean this.

I think they envisage that the constant current could be produced by a neuromodulator in the extracellular fluid. Eg. in the paragraph "In summary, the global phase of a local circuit can be modulated by pulses, such as those mediated by a synchronized volley of synaptic inputs or optogenetic stimulation (Cardin et al., 2009; Sohal et al., 2009), whereas changing the internal phase between local E and I required a constant depolarization to be maintained for as long as the internal phase needs to have the altered value. Such slower time scale modulations could be mediated by neuromodulators (Buia and Tiesinga, 2006)."

When it refers to phase shifts for spikes, it refers to the constant depolarising current as a driving current (if I understood correctly); the wording -to me - suggests a type of 'pacemaker' activity, where the neuron is 'driven' to threshold, fires, hyperpolarises, and depolarises again to threshold, and so on, due to the driving current. However, I have been told that it probably refers to shifting the resting potential closer to threshold and it remains stable at this more depolarised potential (providing there isn't any other excitatory or inhibitory input). Is this more likely what is meant when papers mention a constant depolarising current?

Thanks for the help so far.

 

[atyy]

When it refers to phase shifts for spikes, it refers to the constant depolarising current as a driving current (if I understood correctly); the wording -to me - suggests a type of 'pacemaker' activity, where the neuron is 'driven' to threshold, fires, hyperpolarises, and depolarises again to threshold, and so on, due to the driving current. However, I have been told that it probably refers to shifting the resting potential closer to threshold and it remains stable at this more depolarised potential (providing there isn't any other excitatory or inhibitory input). Is this more likely what is meant when papers mention a constant depolarising current?

The term "driving current" doesn't have a very specific meaning. Its general meaning is a current that causes a neuron to spike. However, the specific use varies from paper to paper. As long as you understand the use of the term within that paper, that's fine.

Take Figure 2 for example. The neuron receives excitation and inhibition governed by periodic processes (consisting of a bunch of periodic Gaussian-like 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 Gaussian-like excitation is also a driving current (or at least a driving input), but in Fig 2 they reserve that term for the constant current.