Periodic Patterns in Recurrent Networks of Varying Neuron Types

The first neuron type we looked at was the Integrate-and-Fire Neuron Model. The mathematical model for an integrate-and-fire neuron, j, is as follows:

where &tau is the membrane time constant, V_rest is the resting potential, V_j is the membrane potential, I_j is the external input, and I_exc and I_inh are the inputs from the excitatory and inhibitory synapses, respectively.
More specifically,

and

where g_exc,j is the excitatory synaptic conductance and g_inh,j is the inhibitory conductance, of neuron j.

The values of g_exc,j and g_inh,j follow the same rule. Whenever neuron i fires (i.e. V_i > V_&theta) not only does V_i get reset to V_reset (where V_reset < V_&theta), but also for any neuron j that i forms a connection with:

for both excitatory and inhibitory synapses.
When no connecting neurons have fired, however, the conductances show exponential decay like so:

where g_0,j is the conductance after the last spike, and &tau_syn is the synaptic time constant (typically much shorter than the membrane time constant). G_i,j is a constant representing the strength of the connection from i to j.

With a network of 100 of these integrate-and-fire neurons, it doesn't take long for a nice pattern to emerge:

Quadratic Neuron

We thought it would be interesting to see if similar patterns would emerge if we used a different neuron model within the same neural network structure. The model we chose was the quadratic neuron:

where U has units of potential (in our case, we set U to be near V_rest). The conductances follow the same rules as in the integrate-and-fire neuron.

Here is an example of a spike raster plot from such a network:

With the quadratic neurons, the networks behaved a bit differently. For one, usually more neurons seemed to fire at least once, but fewer remained firing after a short while, and the ones that were usually had a high firing frequency. Despite running the simulation three times longer than with the integrate-and-fire neurons, I still did not see a consistent pattern emerge. It seems to be very close, but not quite right. There are usually a few neurons that have distinctive patterns in their firing, but even those are a little inconsistent. And then there are always a few neurons randomly firing at a high rate without a discernible pattern. For instance in the above plot, neurons 3, 6, 7, 9 and 10 are never quite consistent. Perhaps making precise changes in some of the parameters of the system would result in the emergence of perfect stable patterns, however none of the changes I tried seemed to do the trick.

Conclusions

While I could not get a stable periodic pattern to appear with perfect repetition with the quadratic neural network, I still think that our results are significant. There are patterns to the firings of certain groups of neurons, to within a certain amount of error. Unfortunately, I think the system as it has been modeled is so sensitive to changes in the parameters that it is very difficult to achieve a set of parameters that behaves ideally. This is where our ability to model nature has broken down, for networks of real neurons are more robust than this. They are not hypersensitive to changes in inputs or conductances, as our model here is. It would still be good to demonstrate perfect stable periodic patterns with a myriad of different neuron models to show that our underlying assumptions about pattern-forming in recurrent networks are correct. However, again, I could not seem to do that given the limitations of my system. Perhaps my results here would be the equivalent of a fuzzy memory or a calculation gone slightly awry in a real brain.

References

[1] Dezhe Z. Jin, "Fast convergence of spike sequences to periodic patterns in recurrent networks", Physical Review Letters, 89, 208102 (2002). (PDF)

[2] Hansel D, Mato G. "Existence and stability of persistent states in large neuronal networks," Physical Review Letters, 86, 4175 (2001).

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Last modified: 5/23/2003