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The Stochastic Nature of Single Neurons
Kamran Diba, Christof Koch

Our labs have been very active in furthering our understanding of the biophysical noise in neocortical pyramidal cells. The Hebrew University group traveled to California in March, and Dr. Kamran Diba traveled twice to Jerusalem in April and August to discuss and advance our collaborative research. Theoretically, we have strengthened our understanding of the role of ion channels and synaptic vesicular release in determining the voltage noise fluctuations. Experimentally, we made more measurements under varied pharmacological conditions. We also developed a method for quantifying instrumental noise, and we began measuring the input impedance of the cell with zap currents. We presented a poster at the Society for Neuroscience meeting in November. We are presently working to understand some of the low-frequency noise features that we recently uncovered.

Figure 1. Various Current Power Spectral Densities per Channel at -65mV per channel current power spectra is in units of mA2/Hz. Not surprisingly, synaptic events lead to the greatest current fluctuations. As for ion channels, transient calcium channels give the largest current fluctuation.

Figure 2. Passive and Apparent Conductances for Various Channel Types (pS). We have assumed a single channel conductance of 20pS for potassium and sodium, and 10pS for calcium and Ih. The apparent conductance, on the right, comes from performing a first order linearization around the holding potential. Its magnitude can be significantly greater than the passive term, shown on the left.

The Model
The effect of ion channels on the voltage noise is two-fold. First, the stochastic nature of channel gating leads to current fluctuations for each channel (and channel type). Second, the channel conductance properties change the response of cell to currents and current fluctuations. Both of these contributions can be modeled and calculated given the cell geometry and the distribution of channels.

Given the kinetic scheme for the gating of an ion channel, the covariance can be calculated based on the theory of Markov chains. By taking the Fourier transform, we arrive at the current spectral power density per channel. Some examples of the single channel SI are shown in Figure 1. To calculate the voltage response due to a current input, we need to compute the transfer impedance Z_trans(x,y,f), which relates the voltage power spectral density at y to the current spectral density at x with

For our calculations, we make use of the “extended” impedance class capabilities of NEURON, which allows us to calculate the transfer impedance using a first order linearization of each conductance about the holding potential, for arbitrary choice of geometry. We can learn much about the relevant channels which are present in a real neuron by looking at how the cell conductance, the D.C. inverse of the impedance, changes with holding potential (see Figure 2).

Real Cells
We performed patch-clamps onto the soma of neocortical pyramidal cells from 13-15 day old rats PN, in current clamp mode. We injected current to set the holding potential. Then we recorded long (2-5 minutes) traces of data. Using the overlapping Welch window method, we calculated the voltage power spectral density at different holding potentials.

Figure 3. Sample impedance responses under the control condition. Impedance increases with depolarization suggesting the presence of transient sodium or calcium channels. Ih may also have a similar effect.

Figure 4. Comparison between control and TTX condition for a sample cell. The noise drops to the level of the most hyperpolarized control condition once TTX is added. This is suggestive that TTX was significantly affecting the response of the cell, but was not itself a major contributor to the current noise. Measurement of the input resistances at the three holding potentials yielded for control, Rin=65M_, 173M_, and 334M_, while for TTX, Ri =74 M_, 83M_, and 57M_. These resistance measurements are consistent with our statements regarding sodium’s effect on the response. It must be noted, however, that for this cell, there is very little synaptic activity visible in all but the most hyperpolarized holding potential! This would suggest that calcium may be giving rise to the fluctuations, rather than EPSP’s.

For all cells investigated, the variance increased with depolarization. Also, significant excitatory background synaptic activity could be observed, suggesting that most of the observed noise comes from synaptic activity. We counted visually identifiable EPSP’s. The observed total excitatory synaptic rates was about 2-5Hz. In some cells, IPSP’s could also be observed.

We measured either the input resistance, by injecting a current pulse, or the input impedance, by injecting a zap current, and analyzing the resulting voltage change over several trials. Sample impedance behavior is shown in Figure 3.

As the holding potential is increased there is a dramatic increase in input resistance, indicating that voltage-gated sodium (and possibly transient calcium) channels dominate the overall conductance of the cell. The fact that the impedance increases with depolarization indicates a weak presence of voltage-gated potassium channels (either delayed-rectifier or KA), relative to sodium and calcium. This follows from the fact that potassium channels tend to decrease the input resistance with depolarization, whereas we observe the opposite, even with the addition of TTX.

To compare with experimental results, we modeled a ball and stick neuron with Hodgkin and Huxley type sodium, and potassium channels, as well as synaptic input. The geometry was chosen to give a physiologically reasonable surface area. The calculated voltage power spectra were comparable to what was observed experimentally. In the lower frequency range, the input impedance is considerably increased with depolarization due to the presence of sodium channels (and possibly transient calcium channels.) However, the excitatory synaptic noise changes only slightly with varying holding potential. Thus most of the change in noise with holding pattern is a result of the change in input impedance.

We investigate this claim by blocking sodium channels with the addition of TTX to the bath solution. In general this leads to a decrease in the overall noise, as expected. However, this is mostly not due to a decrease in spontaneous activity, which appeared to be the same before and after TTX. This indicates that the synaptic activity observed is mainly due to minis, which are not invoked by presynaptic action potentials.

From Figure 1, we don’t expect much noise coming from the stochastic fluctuations of sodium channels. Instead, sodium affects the response of the cell, causing amplification of input current noise when activated.

Figure 5. Comparison between control and TTX condition in a ball and stick model. We used the following parameters: ball area = 1750_m², stick diameter = 15_m, stick length = 1000_m, K⁺ density = .5/ _m², Na⁺ density = 1.8/ _m², single channel conductance = 20pS, EPSP density = .04/ _m² , EPSP firing rate = .01Hz, EPSP g_peak = 1200pS, EPSP t_peak = 1.5ms. These parameters give an overall firing rate of 6Hz which is comparable to that observed physiologically. The ball area and stick diameter were taken to correspond to somatic and dendritic area, with stick length otherwise fixed at 1mm. The sodium and potassium densities are similar to found in pyramidal cells.

Figure 6. Population behavior for control vs. synaptically blocked condition. While there is a slight decrease in the standard deviation when we block synapses, the remaining standard deviations are still very significant (around .25mV). However, sodium and potassium noise from a model (shown in black) with reasonably constrained input impedance values, is much less noisy, giving a standard deviation that is generally closer to .1mV. This represents essentially the maximum noise we could attain with these channels.

We added DNQX and bicuculline to the bath to block the observed minis (Figure 7). Modeling work suggests that if the only sources for noise are voltage-gated potassium, voltage-gated sodium, and spontaneous synaptic input, complete block of the synaptic noise would result in a decrease in noise by a couple of orders of magnitude, resulting in variances on the order of .1mV, essentially reaching the resolution power of the experimental setup. That is to say, as long as we constrain channel densities to remain consistent with observed input impedance values (ranging from ~ 100 M_s to ~ 400 M_s), the noise from the stochastic behavior of the sodium and potassium channels is too small to account for the variances we observe experimentally (see Figure 6). This is strongly suggestive that there is another source for the observed fluctuations. These may be calcium or cation currents. It is largely manifested in the lower frequencies, pointing to slower fluctuations. This noise either comes from a new current arising from a side-effect of our choice of blockers, or it indicates the presence of some other current type, presumably transient calcium. We must perform further experiments using different pharmacological agents, to test these possibilities.

Figure 7. Comparison between control and DNQX + bicuculline condition for a sample cell. The amplitude decreases across most frequencies, with the notable exception of frequencies less than .5 Hz. The overall variances are only somewhat decreased.