9.29, Massachusetts Institute of Technology
Introduction
Many processes essential to life are periodic. Many changes in the environment occur at predictable intervals, and many adaptive behaviors require repetitive motor movement. It seems then that a major problem for a nervous system to solve is the generation of neural activity at precisely regulated (and thus dependably predictive) frequencies. Several strategies for the production and modulation of these internal rhythms present themselves; some involve the complex interaction between many neurons, and some involve rhythmic activity within a single neuron. This project examines one way a single neuron can maintain a certain frequency of activity: by the regulation of the expression of ion channels imbedded in its membrane. The expression of an ion channel is represented by its maximal conductance, and the maximal conductances of the various channels in a Hodgkin-Huxley type model neuron form a basis for an eight dimension parameter space in which regions of qualitatively and quantitatively similar behavior are searched for. Presumably, a neuron can maintain a certain frequency under mutation by coupling the expression of certain ion channels that have opposite effects on the behavior. This project hopes to show that this mode of regulation is possible in a model neuron in which the maximal conductances are free parameters.
Background
Some neurons are known to fire repetitively even in the absence of synaptic input. They are called pace-maker neurons because they drive the activity of surrounding cells. A heavily studied pace-maker neuron is the pyloric dilator of the stomatogastric ganglion. This ganglion controls digestion in the stomach of certain crustaceans. The normal mode of regulation described in this neuron is of the activity-dependent kind, in which the neuron can measure its own firing rate and homeostatically adjust its output. Such mechanisms are certainly useful for the fine control of the neuron's activity, but how could it regulate its firing rate under more dramatic changes, such as mutation, in which the firing rate would drop to zero? To answer this question, we must more closely examine the contribution of the several different ion channels to the neuron's activity.
Table 1
| Species of ion channel: | Leak | Na+ | K+ delayed rectifier | A type K+ | KCa | Ca Transient | Ca Slow | H |
| Reversal Potential: | -50 | 50 | -80 | -80 | -80 | high | high | -20 |
Table 1 lists the reversal potentials for the 8 ion channels under consideration in this model. The reversal potential of an ion channel is the Nernst equilibrium potential of the ion in question, and thus the voltage to which the neuron is driven if that channel is more activated, more deinactivated, or if the maximal conductance is increased. If this reversal potential is higher than the threshold potential, such manipulations should increase the probability of the neuron firing, or decrease it if the reversal potential is lower than the threshold potential. The ion channels are modeled as a population with activation and inactivation variables m and h, which are voltage dependent (and [Ca] dependent in the case of KCa). An increase in the maximal conductance of a channel represents an increase in the population of that type of channel in the membrane. The population of a given ion channel is under genetic control, and thus the population of different channels can be linked through the expression of the genes that build them. We can model the effects of manipulating of the maximal conductances to determine how they influence behavior. The two behaviors in question are the frequency of firing, and the number of action potentials that spike closely together, called a burst (illustrated below).
Figure 1
The Model
The activity of a neuron can be simulated with a computer program that solves the differential equations that govern the voltage-dependent gating variables and time constants for the different channels, the concentration of Calcium, and the membrane potential of the cell (which depends on the gating variables).
Figure 2
The above equations are solved numerically by a C++ program courtesy of Mark Goldman. The program uses a numerical technique called exponential Euler in which, for example, V(t+dt) = Vinf + (V(t) - Vinf)*exp(-dt/tau). All the differential equations in the model are solved similarly with a dynamic dt (min = 0.03, max = 0.1). I wrote a similar program for MATLAB, but did not have time to debug. To see how the simulation works, follow the MATLAB code (and email me if you see the error(s)!).
The Problem
The problem of intrinsic regulation is motivated by an experiment by MacLean et al in which the A type K+ channel of the stomatogastric PD is overexpressed. According the our model, overexpression of this channel should have the effect of reducing the activity of the neuron because the reversal potential for K+ is much lower than the threshold potential (which is why K+ is used to rectify the membrane potential after spikes). The effect of increasing the maximal conductance of the A channel in the model is shown below, with an arbitrary selection of maximal conductances for the other channels. (Each point represents the results of running the simulation for 40 seconds; the program measures the frequency and counts the spikes per burst).
Figure 3
According to MacLean et al, the current through the A channel was observed around double its normal value. In our model, this would have the effect of silencing the neuron, but the activity of the actual neuron was unchanged. How can we explain this finding? MacLean also observed a proportional increase in the current through the H channel (constant of proportionality around 40), whose resting potential is above the threshold of the neuron (see figure below). (The H channel is a nonspecific ion channel that lets through a variety of positively charged species). These two observations suggest a mechanism of intrinsic modulation. Since they have opposite effects on the tendency for the neuron to fire, if their expressions are linked, then damage to one could be compensated for by proportional expression of the other. This plausibility of this hypothesis is tested here. Under manipulation of the maximal conductances in a similar manner as proposed by MacLean et al (double A channel, increase H channel by a factor of 40), if the neuron's activity pattern doesn't change, then the hypothesis is not disconfirmed. If such a manipulation of H does not restore the neuron's activity, then perhaps a more complex explanation is required.
Figure 4
(The only curiosity in the
above graph is the counterintuitive effect on the number of spikes per burst. This will come up again later.)
Method and Results
The activity of the model neuron under these manipulations was observed in the computer simulation. The parameters were adjusted by hand after each 40 second simulation. (An alternate approach is to automate the parameter adjustment; I wrote a MATLAB program that feeds the maximal conductances into neuron.m and computes the crosscorrelation between the generated spike train and the spike train with the unmanipulated A and H maximal conductances. The cross correlation is computed for each value of H in a specified range (with no lag), and the best fit is returned. This approach could be generalized to all 8 dimensions of the parameter space to systematically explore the behavior in all regions, though it would take a long time. The code for this alternate approach is available here. Unfortunately it could not be implemented because of the failure of neuron.m to properly simulate the neuron.). The results of these multiple simulations are shown below.
Figure 5
Note the partial success. The H-current adjusted neuron fired at the correct frequency (2.1 Hz) when increased by a factor of 4, but the bursting behavior is not present. (Why this behavior was regenerated after magnification of the H channel by a factor of 4 instead of 40, as would be predicted from MacLean et al is a curiosity that could be further investigated in different regions of the parameter space, but to which I have no answer.) This result is partly understandable as a consequence of the effect on bursting we observed earlier from increasing H, namely that the spikes per burst decreased. Why the H current has this effect on bursting would be an interesting topic for further investigation.
We know that the bursting behavior is strong influenced by the intracellular Calcium concentration, and that this relationship is somewhat complex. Shown below is the effect on bursting due to the increased expression of transient Calcium channels.
Figure 6
Note the nonlinear effect that the transient Calcium channel has on bursting behavior. The spikes per burst is maximal at gmaxCaT=2.5 mS/cm^2 and after gmaxCaT=4, the neuron is nonbursting. This cutoff value for gmaxCaT held for a range of values of gmaxCaP, which indicates that the transient Calcium channel exhibits peculiar properties that are not sensitive to the CaP. Unfortunately, time constraints did not allow these cutoff values of CaT to be tested along other dimensions in the parameter space, but this initial finding demonstrates the complexity of the forces that generate the bursts.
Given the importance of the intracellular Calcium on bursting, one possible solution presents itself (actually, it was presented by Mark Goldman). The bursting curve of Figure 5 shows a rise between bursts that the lower curve does not. This slower rise in the membrane voltage could be due to an effect from the Calcium dependent Potassium channel. If the KCa channel was too sensitive to changes in Calcium concentration (or if the changes that occurred were magnified by too much expression of the channel), then potential bursts would be "rectified" before they could ever fire. If this phenomenon is indeed happening here, that could help explain the reluctance of the boosted H channel neuron to burst. Figure 5 shows a small hump after the single spike of the H-channel compensated neuron, as if it wants to burst, but is being inhibited by K+. To investigate this possibility, I ran the simulation again with KCa decreased from 19 mS/cm^2 to 13. This had no noticeable effect on the wild type PD neuron simulation, but it did restore the bursting capacity of the A and H channel mutant. Figure 7 below illustrates this effect.
Figure 7
Discussion
Though not fully restored to the original state of firing, this demonstrates the restoration of qualitatively similar behavior is possible through the simple linking of the expression of ion channels with opposite effects. The fact that the magnitude of the effects of the different channels is different leads to two conclusions: one, that the proportional increase of one channel (H in this case) must be must larger than the more effective channel (A in this case), and two, that the complexities of the internal struggle between channels for dominance of the membrane potential makes this relationship valid during only certain regions of the parameter space. Given this initial success, further work in this area should more systematically investigate the sensitivity of this mechanism of regulation along different dimensions of the space, particularly with respect to Calcium dependent channels. The degree to which a neuron can maintain its intrinsic properties in the face of genetic mutation, of the A channel in this case, will be represented by the size of the region of insensitivity to other channels. In other words, if the region is small, then only a few neurons (who have been selected to fire at frequencies defined by the parameter values in the region) will be capable of this intrinsic regulation mechanism. If the space is large, and if many different kinds of neurons can be shown to exhibit regions of stability in their parameter spaces, then this instrinsic regulation mechanism will have been demonstrated to be quite robust. The limitations that will be elucidated by this investigation will be a starting point for explaining why other, extrinsic, methods of regulation are necessary to maintain certain patterns of firing.
Goldman, M., Golowasch, J., Marder, E., and Abbott, L.F. Global Structure, Robustness, and Modulation of Neuronal Models. The Journal of Neuroscience, 21(14).
MacLean, J., Zhang, Y., Johnson, B., and Harris-Warrick, R. Activity-Independent Homeostasis in Rhythmically Active Neurons.Neuron, Vol. 37,109-120
Liu, Z., Golowasch, J., Marder, E., and Abbott, L.F. A Model Neuron with Activity-Dependent Conductances Regulated by Multiple Calcium Sensors. The Journal of Neuroscience, 18(7).