Synaptic unreliability

Cortical excitatory synapses are highly noisy. In the experiment shown inFigure 6.2,

five closely-spaced action potentials followed by a single delayed one, are delivered to a presynaptic pyramidal cell, and the EPSP responses are recorded in a postsy- naptic cell. Although a depressing trend is evident in the ensemble average (bottom trace), the responses from spike to spike, and from trial to trial, are seen to be highly variable in amplitude. There are numerous failures to release transmitter in response to presynaptic action potentials, although estimates of the overall probability of re- lease vary greatly [13, 23, 26].

The distribution of amplitudes of response at a given synapse is wide – when responses to single presynaptic APs, separated by at least several seconds, are mea- sured, there is evidence for a quantal or multimodal distribution of amplitudes [16]. However, whatever this distribution of amplitudes is, it is not uniform in time during a sequence of synaptic responses. For example, immediately following a release,

162 Computational Neuroscience: A Comprehensive Approach

5mV 2mV 500ms

Figure 6.2

An ensemble measurement of EPSPs in a layer 2/3 pyramidal neuron. Top 6 traces: individual trials. Red trace, ensemble average. Dotted lines indicate times at which presynaptic neuron is stimulated. Recording by Ingo Kleppe, Dept. of Physiology, University of Cambridge. (See color insert.)

there is an increased probability of failure, and vice versa [59]. In other words, there are significant correlations over time in the variability at individual synapses, as in- deed there are in the mean synaptic response. These have been shown to extend over surprisingly long time scales, of several minutes or more during natural-like spike trains [35]. Further understanding of the variability or reliability of individual synapses over time during complex, natural input, is an important goal. One might think that the effect of synaptic fluctuations will be averaged away because there are large numbers of synapses per cell. However even if many inputs are active, only one completes the job of taking the voltage over threshold. As will be discussed fur- ther below, the nonlinear threshold behaviour of neurons means that the properties of essentially any input fluctuations, however small, can determine the firing pattern of the cell – it all depends on the proximity of the spike generation mechanism to its threshold.

The Biophysical Basis of Biring Variability in Cortical Neurons 163

6.4

Postsynaptic ion channel noise

The variability introduced by ion channel gating in the postsynaptic cell is rather bet- ter understood than synaptic unreliability (see chapter by Hawkes). Single-channel recording has shown in great detail how single ion channels operate as probabilistic machines, or Markov processes with a reasonably small number of discrete states, corresponding to distinct conformations of the ion channel protein [29]. Single chan- nel properties of ion channels – their conductance and average opening duration – were first estimated, before the advent of the gigaohm-seal patch-clamp, by analysing the current or voltage noise produced by channel gating. The properties of this noise have therefore been well characterised and extensively measured [11]. The effects of ion channel gating noise in neurons have recently been reviewed in [66].

If channels in an identical population of size N are independent, and have a single conducting current level i then at steady-state, then the population current variance

σ2_{= iI − I}2_{/N, where I is the mean population or macroscopic current. The size of}

fluctuations therefore scales with the square root of the size of the population cur- rent, for small open channel probability. For large numbers of channels, the noise amplitude is distributed normally. How big are the populations of channels? This question is complicated by the fact that they are distributed over large electrical dis- tances within the cell. It is the local population of channels, in particular around any site of AP initiation, which is relevant. The density of sodium channels in layer 5 pyramidal neurons appears to be fairly uniform in the soma and dendrites, at around several channels per square micrometre, but is probably much higher in the proximal axon, where it is thought that many action potentials initiate [61]. Both calcium and sodium action potentials can, however, be initiated in the remote dendrites. There are several types of potassium channels, the density of which tapers off with dis- tance from the soma in pyramidal neurons [36], but is probably in the region of 0.1 to 1 channels per square micrometer. Less is known about the density of functional calcium channels in cortical pyramidal neurons, but in hippocampal pyramidal neu- rons densities of 1 to 10 channels per square micrometer of high and low voltage activated calcium channels [40]. The noise from persistent Na channels, because of their maintained activation around threshold, appears to be particularly important in controlling firing patterns in entorhinal cortical cells [65]. However, it is important to realise that the population of open channels is in general much smaller than the total. Channel noise becomes most powerful in its effects when only a very small proportion of channels are open [51]. This is because the size of fluctuations relative to the mean conductance is highest under these conditions.

Another important quantity in determining the variance of channel noise is the size of the single channel conductance. Amongst voltage-gated channels, this varies from a few pS for a low voltage-activated calcium channel to about 200 pS for a maxi calcium-activated potassium channel. At the excitatory synapse, the AMPA receptor has a single channel conductance of around 10 pS [63], while the NMDA receptor has a conductance of 35 to 50 pS [47].

164 Computational Neuroscience: A Comprehensive Approach

Figure 6.3

Spectrum of NMDA receptor current noise at a single synapse. The spectrum follows a sum of two Lorentzian components, with corner frequencies of 25 Hz and 110 Hz. From [47] with permission.

Equally important, though, is the timing, or frequency distribution of the fluc- tuations. The membrane time “constant” (which is actually dynamically-varying) determines the low-pass filtering of the input signal. Low frequencies in the noise e.g. below 100 Hz are much more effective in distorting the membrane potential than higher frequency fluctuations. Under stationary conditions, channel noise is essentially a linear stochastic process: its autocorrelation function (or equivalently power spectrum) contains all the information available to predict its time course. The autocorrelation function is a sum of exponential components whose rates are given by the eigenvalues of the kinetic matrix (see chapter by Hawkes). Correspondingly, the power spectrum is a sum of Lorentzian components. Almost always, though, for actual channels, the power of one or two of these components is dominant. For

example,Figure 6.3shows the noise through NMDA receptor channels at a single

synapse [47]. The power of the lower frequency component is more than ten times that of the higher frequency component.

Roughly speaking, this first component corresponds to the correlation due to the mean open lifetime of channels or predominant burst lifetime. For purposes of mod- elling the function of this noise, therefore, it is often enough to consider a single exponentially-correlated process. A continuous stochastic process which has an ex-

ponential autocorrelation, with amplitudeσ and correlation time constantτ, is the

Ornstein-Uhlenbeck (OU) process [18, 24],ξ(t), which can be generated by numer-

ically solving:

τdξ

dt =

−ξ

The Biophysical Basis of Biring Variability in Cortical Neurons 165

where gw(t) is Gaussian white noise, and the standard deviation ofξ is

cτ/2, and c is a constant.

This generates stationary noise. However, real channel noise is usually not sta- tionary: for example, transmitter-activated channels have transition rates which de- pend on the changing transmitter concentration, and voltage-activated channels have voltage-dependent transition rates. To account for such nonstationarity with high accuracy, it is necessary to use stochastic simulations of populations of channels, modelling the state transitions of all channels in each population [9, 55]. One may

also use an OU process which is nonstationary inσandτto approximate the stochas-

tic Hodgkin-Huxley system [19]. However, since as described later, a great deal of spike-time variance is generated by noise in a limited band of membrane potential – around threshold – then a stationary OU noise source or sources may be a reasonably accurate yet simple model for predicting firing variability. An OU process is also a good model for synaptic noise, composed of large numbers of small, identical EPSPs or IPSPs which have a fast onset and decay exponentially and whose arrival times are a Poisson process [1].

6.5

Integration of a transient input by cortical neu-