Performance of different local measures of spike time variability and estimating internal state dynamics of cortical neurons.
Adrian Ponce Alvarez1 Kosuke Hamaguchi2,3 Nicolas Brunel2 Alexa Riehle1
1INCM-UniversiteAix-Marseille II, France, 2LPN-Universite Rene Descartes, France, 2RIKEN BSI, Japan

Irregularity of spike timing can be measured by the coefficient of variation (CV), which is a global measure defined as the dispersion of the inter-spike intervals (ISI). However, the CV undergoes large changes by rate fluctuations. Several local and relatively rate-independent measures of this irregularity are found in the literature. Unfortunately, as far as we know, these new measurements have not been compared to each other. Here we present a comparison of four of them: the "local coefficient of variation" named CV2 [1], the local variation LV [2], the metric IR [3] and the metric SI [4]. The LV measure is thought to be useful for cell classification [2]. SI is based on the best estimating function (asymptotically) of the shape parameter in a gamma-distributed assumption of spiking activity without estimating the time-dependent rate, which is an unknown function. The question arises, which of these coefficients is the most efficient for analysing experimental data where the number of inter-spike intervals is limited. The classical CV of cortical neurons usually falls in the range between 0.2 and 2.

We first compared the performance of these measures by simulating point processes with gamma-distributed intervals and using time sliding windows as used in experimental data analysis. We tested the resistance of these measures to non-stationarities and discontinuities of the firing rate. By calculating the distributions of the individual local values we showed that the measure CV2 is the less sensitive to the shortage of samples: it has the lowest estimation bias. In addition, one can calculate analytically the value of CV2 for any gamma-process.

The irregularity of the spiking activity of neurons recorded by A. Riehle and collaborators was analysed using CV2. The neurons from the motor cortex of a Rhesus monkey were recorded during a Choice Reaction-Time task [5]. In this task the animal was trained to perform arm movements in opposite directions. On a vertical panel, three touch sensitive targets were mounted in a horizontal line. Both lateral targets could be lit either in red or green, whereas the central target was lit in yellow. The animal had to initialize the trial by holding the central target. After a delay, a preparatory signal (PS) was presented: both peripheral targets were presented simultaneously, one in red and the other in green. An auditory response signal (RS) then followed after either 600 or 1200 ms. The animal had been trained to associate to each colour one of these two possible delay durations.

Interestingly, the distribution of CV2 statistics of the whole population of neurons calculated from the delay period in the Choice-reaction task shows a clear bi-modal distribution.

Although the firing rate is highly modulated during the task, the CV2 is essentially constant in time. The fact that CV2 remains constant is likely to constrain the architecture of the network in which the recorded neurons are embedded. In fact, if the increase in rate is due to purely excitatory feedforward inputs, from the analysis of simple integrate-and-fire (LIF) type models one would expect the CV2 to decrease as the rate increases. Thus, the basic observation that the CV2 remains constant as the rate increases seems to indicate that this rate increase is at least partly due to recurrent inputs.

Using a simple LIF model one can compute the mean, m, and fluctuations, s, of the inputs that produces a given firing rate and CV2. Then, we can plot contour lines of both firing rate and CV on the m/s plane, and superimpose single neuron data on that contour plot. This tells us how the statistics of that particular neuron changes during the task, if we assume that the neuron can be well approximated by a LIF neuron. The next step is to take a recurrent network model with specific parameters, and compute how the firing rate/CV of neurons in such networks vary as a function of external inputs. This gives a trajectory in the m/s plane, that depends in a pronounced fashion on the excitation-inhibition balance. This analysis suggests that most of the neurons would be operating in an inhibition-dominated network (for many neurons m is far below the threshold although their firing rate is 10~40 Hz).

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