Based on a brief literature review, it appears that Izhikevich's numerical code for neuron simulation has nonzero traction. (For instance, Kramer's simulated neural data is based on the Izhikevich model). More information about the Izhikevich model can be obtained here. Izhikevich seems to be an interesting guy, and I'm eager to find out more about his work.
On the other hand, in the longer run, it's probably good to test the network inference framework on multiple simulation models, so that we are not misled by idiosyncrasies of a particular numerical model.
A first stab at generating a few Izhikevich neurons:
Now, to get at the model in a bit more detail, I would like to see the traces in a more controlled setting: evolution of two neurons where neuron 1 has an excitatory synapse on neuron 2. Here's a "first pass" simulation of two neurons with a specific nonzero coupling between them. I also plot the sample cross-correlation between the two temporal traces:
Here are some thoughts on Izhikevich's numerical model:
- The paper that describes the model is pretty sparse on "biological intuition."
- It seems to me that the >100 mV amplitude spiking that I see in my "local implementation" (which is basically copied and pasted from the Izhikevich paper) is a numerical glitch. Two observations: (1) I find that if I step the numerical model at timesteps other than 0.5 ms (actually, using finer steps), the numerical trajectories become divergent; (2) The bottom panel of Fig. 3 in the paper shows a pretty regular sized amplitude train, which appears to be on the order of <100 mV based on the noise characteristics.
I think getting a handle on the numerical simulation will be important. I suspect that the pulse amplitudes -- which suffer from numeric noise (as I claim above) -- is throwing off the cross-correlation metric.
Another possibility is to use the "digital" firing traces (i.e. did a cell fire at a particular time or not) instead of the "analog" (voltage) time traces.
Next, my observations on how cell-to-cell connectivity is implemented.
Numerical implementation of synapses in the Izhikevich model:
In the Izhikevich numerical model, each neuron has two state variables $(u, v)$ that proceed according to the following differential equation:
- $v' = 0.04 v^2 + 5v + 140 -u + I$
- $u' = a\cdot (bv - u)$
where $v$ represents the membrane potential of that neuron. In the DE for $v'$, the term $I$ is a forcing term that represents the inputs to that neuron. In particular, in Izhikevich's example code the forcing term consists of two terms: a random term (denoted "thalamic input", so presumably we're modeling cortical neurons), and another term that derives from the connection matrix $S_{ij}$.
How to scale the connection strength $S_{ij}$?
In the numerical model, we can set any arbitrary value for $S_{ij}$. Clearly, our ability to distinguish the cross-correlation between two neurons will be aided by choosing a large value of the connection strength. However, it is important to use values for $S_{ij}$ that is biologically relevant.
Unfortunately, the fact that the differential equation is given in "dimensionless units" (even though v is supposed to represent voltage at mV scale and time is ms scale) and the lack of biological insights in the paper makes it hard to determine the reasonable range of $S_{ij}$ values.
Instead, I propose to perform the following experiment. I will hook up a two neuron system where neuron 2 (N2) drives neuron 1 (N1) through a coupling term $S_{1,2}=\alpha$. I will remove the "thalamic" input from N1, so that N1 is driven exclusively by N2 (otherwise, it is difficult to distinguish the effect of N2). I will characterize $\alpha$ by the amplitude of the excitatory post-synaptic potential (EPSP).
Based on this simulation, I find that the (positive) magnitude of $alpha$ is roughly the amplitude of the EPSP. According to Wikipedia, the amplitude of PSPs can be "as low as 0.4 mV to as high as 20 mV". With $S_{1,2}=20$, action potentials from N2 is able to drive action potentials in N1. So, it appears that a range of $S_{ij} \leq 20$ should be biologically feasible.
How to scale the connection strength $S_{ij}$?
In the numerical model, we can set any arbitrary value for $S_{ij}$. Clearly, our ability to distinguish the cross-correlation between two neurons will be aided by choosing a large value of the connection strength. However, it is important to use values for $S_{ij}$ that is biologically relevant.
Unfortunately, the fact that the differential equation is given in "dimensionless units" (even though v is supposed to represent voltage at mV scale and time is ms scale) and the lack of biological insights in the paper makes it hard to determine the reasonable range of $S_{ij}$ values.
Instead, I propose to perform the following experiment. I will hook up a two neuron system where neuron 2 (N2) drives neuron 1 (N1) through a coupling term $S_{1,2}=\alpha$. I will remove the "thalamic" input from N1, so that N1 is driven exclusively by N2 (otherwise, it is difficult to distinguish the effect of N2). I will characterize $\alpha$ by the amplitude of the excitatory post-synaptic potential (EPSP).
Based on this simulation, I find that the (positive) magnitude of $alpha$ is roughly the amplitude of the EPSP. According to Wikipedia, the amplitude of PSPs can be "as low as 0.4 mV to as high as 20 mV". With $S_{1,2}=20$, action potentials from N2 is able to drive action potentials in N1. So, it appears that a range of $S_{ij} \leq 20$ should be biologically feasible.
No comments:
Post a Comment