However, the stochastic nature of neuronal activity leads to severe biases in the estimation of these oscillations in single unit spike trains .

Different biological and experimental factors cause the spike train to differentially reflect its underlying oscillatory rate function.

estimationWe introduce a novel objective measure, the "modulation index", which overcomes these biases, and enables reliable detection of oscillations from spike trains and a direct estima-tion of the oscillation magnitude.

In this manuscript, we expose major biases and distortions which arise from the quantification of neuronal spike train oscillations.

detectionNext, following a formulation of the distortions, we introduce a novel objective measure, the "modulation index", which overcomes these biases, and enables a reliable de-tection of oscillations from spike trains and a direct estimation of the oscillation magnitude.

Analysis of the power spectrum is a common method for identifying enhanced (or reduced) oscillations in neuronal data, and is widely used on a variety of brain signals spanning multiple orders of magnitude, such as electroencephalograms (EEG), local field potentials (LFP), multiunit activity (MUA), single unit spike trains and cellular membrane potentials [5—8].

The time of occurrence of action potentials emitted by a single neuron; i.e., single unit spike trains , are a major source of neurophysiological data stemming from both intracellular and extracellular recordings.

These neuronal spike trains may be viewed as a stochastic point process where a discrete event represents each action potential [9].

1—2 is linear with respect to the underlying spike trains {Ty(1‘)}.

Raw experimental data 3 Stimulus—a varying scalar value on each trial so Threshold stimulus value in the 2AFC task 0* Animal choice—binary report on each trial r,-(t) Spike train from neuron i in a given trial 02 Stimulus variance across trials

We describe the activity of this neural population on every trial as a multivariate point process r(t) = {Ty(1‘)},- = 1. , N“, where each r,(t) is the spike train for neuron i, and Ntot denotes the full population size, a very large and unknown number.

As is common in electrophysiological recordings, we will quantify the raw spike trains by their first and second order statistics.

Classic measures in decision-making experiments can be interpreted as estimates of the first-and second-order statistics of choice c and recorded spike trains ri(t), across all trials with a fixed stimulus value 5:

In practice, they are estimated by binning spike trains ri(t) with a finite temporal precision, depending on the amount of data available.

In addition, the neurometric sensitivities also depend on the time scale w that is used to integrate each neuron’s spike train in a given trial [3, 11—13].

Since the linear readout relies on the integrated spike trains , eq.

4—6, we find statistical quantities related to the percept E. On the right-hand sides of these equations, we find the model’s predictions, which are based on the neurons’ (measurable) response statistics, 19 and C. More specifically, the first line describes the average dependency of § on stimulus s, the second line expresses the resulting variance for the percept, and the third line expresses the linear covariance between each neuron’s spike train , and the animal’s percept § on the trial.

If the feedback depends linearly on per-cept§ (and thus, on the spike trains ), its effects are fully encompassed in our model.

One way around these difficulties may be to use large-scale extracellular recording of spike trains and apply statistical methods to model and infer functional connections between neurons.

To determine whether an input of a certain amplitude can be “detected” given a specific set of spike trains we use the log likelihood ratio (LLR).

Both the history-only Model 1 and Model 2 with constant coupling capture statistics of the spike trains and predict spikes with reasonable accuracy.

On the other hand, methods for recording extracellular spike trains are advancing at a rapid pace [9,10] and allowing the simultaneous recording of hundreds of neurons.

Estimation of synaptic interactions from extracellularly recorded spike trains requires development of sensitive data analysis tools.

Although strong synapses are usually readily detectable using cross-correlation analysis [11—17], where they appear as asymmetric, short latency peaks on cross-correlograms [18,19], in general, it is difficult to link the statistical relationships between spike trains to specific synaptic processes [20,21].

Statistical inference of synaptic connections of different strength from spike trains .

D) Detectability of synaptic connections from spike trains : Dependence of the log likelihood ratio between Models M1 and M2 on the input amplitude.

Detectability of synaptic connections from spike trains depends strongly on how much data is available.

Comparison of the distributions obtained using the observed vs shuffled spike trains allows us to test whether an input has a statistically significant effect on the firing of the postsynaptic neuron (see Methods).

To separately control the correlations within and between the pre-synaptic pools of the striatal neurons, we extended the multiple-interaction process (MIP) model of correlated ensemble of Poisson type spike trains [12, 23].

The MIP model generates correlations by copying spikes from a spike train (the mother spike train) with a fixed probability (the copy probability, which determined the resulting correlation) to the individual spike trains .

By making many convergent connections using the ‘lossy synapse’ we can mimic the random copying of spikes from the mother spike train to the children process.

The spike trains in each pre-synaptic pool are themselves correlated With a correlation coefficient W. For such pooled random variables, Bedenbaugh and Gerstein [22] derived the following relationship: where p12 is the correlation coefficient between the two pools of the pre-synaptic neurons, 11 is the size of each pre-synaptic pool.

Intuitively we can understand this relationship between B and W in the following way: Imagine two pools of identical spike trains, but with individual spikes trains uncorrelated to each other (i.e.

In this case, the average correlations between the two pools will be small because each spike train has only copy in the other pool, while being uncorrelated with all others.

The empirical Bernoulli information is strictly greater than the estimated single-spike (or “Poisson”) information for a binary spike train that is not all zeros or ones, since To > 0 and these spike absences are neglected by the single-spike information measure.

Quantifying MID information loss for binary spike trains .

Thus, for example, if 20% of the bins in a binary spike train contain a spike, the standard MID estimator will necessarily neglect at least 10% of the total mutual information.

However, real spike trains may eXhibit more or less variability than a Poisson process [fl].

Despite the diversity and variability of input spike trains , neurons can learn and represent specific information during developmental processes and according to specific task requirements.

In reality, however, there would be crosstalk noise among input spike trains caused by the interference of external sources.

In the model used by those authors, the synaptic weight matrix is treated as a hyper parameter and estimated by considering the maximum likelihood estimation of input spike trains .