This week we continued last week's discussion by focussing in on some of the details of modeling neuronal spiking using a computer. The gold-standard model is Hodgkin Huxley - although we'd talked about it last week, we worked in more detail this week to understand how the Markov model of subunit state leads to a first order differential equation describing the dynamics of opening and closing membrane proteins. For example, suppose we are discussing the "n" gates comprising the potassium channel. Suppose "n" percent of them are open and (1-n) percent are closed. The probability of transitioning from closed to open in any instant is "alpha" and the probability of transitioning from open back to closed is "beta". The subunit dynamics can be described by dn/dt = (1-n)*alpha - n*beta. This states that the change in percentage of open "n" subunits is the number of closed subunits times the probability that each one will open, minus the number of open subunits time the probability that each one will close. If we solve dn/dt numerically, we will come up with a function n(t) that describes the proportion of n subunits open as a function of time. The potassium conductance is then given by gk_max * n^4, where the n^4 indicates that each K+ ion channel has four n subunits and all four must be open for the channel to conduct. It turns out that the dn/dt equation is just a first order differential equation with time constant 1/(alpha + beta) and n_infinity given by alpha / (alpha + beta). What makes Hodgkin Huxley a real challenge is that alpha and beta are both functions of membrane voltage!
Afterwards, we discussed simpler models that are easier and faster to implement that don't sacrifice too much precision. The basic model is integrate-and-fire. We looked at how to model synaptic input into an IAF neuron: essentially we extend our ion channel model to include an ion channel that opens and closes according to some post-synaptic conductance model, of which there are several. Using this configuration, it is easy to construct arbitrarily large networks of neurons that synapse onto each other and exhibit any number of behaviors.
Next we talked about the Izhikevitch neuron model, which is an integrate-and-fire type neuron but that includes two differential equations. The first models membrane voltage and the 2nd models membrane inactivation; the two equations work in opposition so that the more the membrane voltage increases, the more the inactivation increases to counter it. The dv/dt equation includes a quadratic equation that has been fitted based on empirical measurements from live tissue. The beauty of Izhikevitch is that it includes four parameters that can be varied to produce all sorts of different neural spiking patterns that actually occur in real life. In this way it is possible to build a network of neurons with an impressively rich palate of properties; the synaptic models (previous paragraph) can be used to interconnect these neurons.
Finally, we had an impromptu mini-lecture from one of our Neuroscience grad students who taught us about how G-protein modulated ion channel up-regulation is responsible for late phase synapse potentiation. This is something I know very little about and was very interested to learn! The Wikipedia entry on Long Term Potentiation covers this subject nicely.