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In this post we look at those neuron models whose steady state voltage plot has a single spike. In PBM they observed that plotting spiking neurons by frequency yielded a bimodal distribution with two essentially separate distributions. Also, in PBM they separate tonic spikers from 1 spike bursters using an area calculation, in exp3 we did not calculate that area feature so we cannot separate tonic spikere from one spike bursters. For this post we will call these single spike neurons they can be tonic spikers or bursters.

Given that understanding, we can plot the periods of all single spike neurons as shown below and we see that there are several peaks



and we can use these peaks to define several domains of single spike neuron models, classified by their periods:

D1: period in 0.05-0.10 sec with peak at 0.016 sec
D2: period in 0.10-0.38 sec with peak at 0.30 sec
D3: period in 0.38-0.78 sec with peak at 0.50 sec
D4: period in 0.78-3.00 sec with peak at 0.92 sec

The following colormap colors neurons in these domains using red, green, yellow, and purple respectively:



and here is an ndvis view of this colormap with projection kca-na-cas-h for x and cat-kd-a-leak for y



In this plot the four domains D1,D2,D3,D4 all seem to map to spatial separate regions which we can call R1,R2,R3,R4 in this post. You may need to click on the image to get a larger version where it is easier to see these four regions.

Observe that D1 is concentrated in the region roughly given by
KCa<2 Kd>=1, Na>=1, and as KCa increases it tends to thin out where Na+Kd is low
Moreover there is a tendency for D1 to be in the lower right corners of the Na*Kd cells.

The D2 region appears only when CaT=0 and where Kd>Na (i.e the upper left corner of the level 2 cells, that is the Na*Kd cells)

The D3 and D4 regions satisfy
KCa>=1, (Na<1 or Kd<1 or Na+Kd<2)
to get a better way of separating these regions we need to look at another projection.

Below is the image for the projection with x= na-cas-kca-h and y=kd-cat-a-leak



Observe that the D4 region mainly satisfies CaS<2, while the D3 region has CaS>2.

Although we haven't tried to verify it yet, the correspondence between these domains and the PBM domains seems to be roughly the following:

D1: fast spikers of PBM
D2: slow spikers of PBM
D3 and D4: one spike bursters of PBM

In a later post we will investigate methods for finding hyperplane equations for the boundaries of these regions.

We have seen in the previous post that the distribution of membrane potentials among silent neurons sampled using a random perturbation around regular grid is a multimodal distribution, shown below:



There are two main peaks: the tall peak centered around -50mV and the broad bimodal region from about -25mV to -10mV. There is also a small bump, only visible in the log plot (in red) from about -40mV to -25mV. Looking more carefully, we can refine these boundary potentials to -18mv, -31mV, and -42mV and use this to define a color map.

Here is the color map we use (move the slider to view the colors at the far right):



This example also shows how we use convert perturbed grid data into ndvis coordinates, a better approach would be to precompute the integer coordinates (using round) and use those to create a model number which we could index over. This would allow for more rapid access to the data since we wouldn't have to call "round" eight times for each pixel.

Here is the ndvis image for this color map with the "standard" projection na-cat-cas-a for x and kca-kd-h-leak for y:



Some Observations
One thing we see immediately from this picture is that the four regions which were defined by their membrane potential ranges all lie in different regions of parameters space. Lets let D1,D2,D3,D4 represent the four different kinds of silent neurons we have identified according to their membrane potential:

• D1: -18mV to 2mV, orange


• D2: -31mV to -18mV, blue


• D3: -42mV to -31mV, aqua


• D4: -63mV to -42mV, green

These domains are defined by neuron model properties. What we have observed is that these domains also map nicely to four regions:

The orange domain D1 (-18mV to 2mV) is concentrated in the region R1 which is the bottom row of each cell in the bottom row of the image, that is, in the region KCa=Kd=0 and it seems a little more dense in the subregions where CaT is large.

The blue domain (-31mV to -18mV) is in the region R2 which is mainly in the lower left cell, where Na=KCa=0, but it seems to extend with linear drop off into NA=1,KCa=0 and NA=1,KCa=1 but concentrated in all cases in the subregion where CaT >=3.

Observe that the blue and orange domains seem to intermix in the intersection of their corresponding regions (R1 and R2) - the two cells on the bottom left with the blue encroaching on the orange region. These two domains are also not cleanly separated in their voltage distributions as they are the two halves of a bimodal distribution. This would lead one to hypothesize that the perhaps the domains ought to defined by a combination of neuron properties and parameter properties. Perhaps there is something fundamentally different about the silent neurons in regions R1 and R2.

The small aqua region is concentrated in the region R3 which is in the lower left corner (Na=KCa=0) and in that
grid cell it is mostly in the upper right (Kd large, CaT small).

The green region can be better visualized with the following dimension order: cas-a-na-h for x and cat-leak-kd-kca for y, as seen in the following image:



This clearly has a "linear" appearance to is and in a later post we will derive a formula for the boundary as a simple hyperplane.

As a final note, let us compare the image we have just analyzed with the image we get when we apply a similar color map to the data from exp1 which used a non-perturbed regularly spaced grid. The non-perturbed data has more peaks in its distribution and these are mostly represented by the red pixels which don't appear in the image from exp3.



Observe that we have roughly the same regions (the green, blue, and orange) which are roughly in the same locations but the perturbed sampling produces an image with a wider range of densities. The non-perturbed has larger solid color bars. We can interpret the densities in the perturbed sampling image as indicating how many neurons have the property of interest in a particular interval represented by that grid cell. If the density is low then that might mean that the region is concentrated in only one small part of the paramter space represented by that grid.

In future posts we will try to find more precise formulas defining the regions R1,R2,R3,R4 of parameter space that correspond to the domains D1,D2,D3,D4 of silent neurson classified by ranges of membrane potentials. We will also look at the behavior of active neurons near the boundaries of these regions.

Lets now take a look at the boundary between the silent and active neurons in the PBM model using the two data sets discussed earlier: exp1 with a straight grid sampling method and exp3 with a randomly perturbed grid sampling method.

The silent neurons are characterized by one paramter, their resting membrane potential, and this ranges between -100mV and +100mV. Lets look at the distribution plots we get from the two experiments.

First the plot from the fixed grid experiment exp1 is below:



This plot shows a multi-modal distribution with four large regions and two smaller ones at the left. The plot was constructed by grouping neuron models in to bins of size 1mV based on their steadystate membrane potential. The green plot shows the number of neuron models with that potential. The red plot is the log of the distribution function (multiplied by 1000 to fit on the same scale).

Next lets look at the distribution obtained from the random perturbation of parameters around the grid points:



This was created in the same way as the exp1 plot but using the exp3 data. Note that it is a much simpler distribution with only two major modes. The leftmost peak corresponds to a similar peak in the exp1 data, while the rightmost peak corresponds roughly to two exp1 peaks.

The exp3 distribution represents a randomly chosen model but with a distribution that places double the probability on neurons that are within 0.5 of zero in a parameter.

In our book Chapter on silent neurons using the PBM data we performed a detailed analysis of the silent neuron regions based on the strict grid sampling and we identified six distinct regions of the parameter space that corresponded the six main peaks in the distribution. We will not go into that data in detail except to note that all but two of those regions disappear in this new sampling.
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