See Neuronal network

Cross-correlation

A quantity that is proportional to the synaptic transmission curve, and which can be readily experimentally and computationally determined.

See definition of cross-correlation here

The cross-correlation between spontaneously active nerve cells will be affected by the structure of the presynaptic spike train. Mathematically, the only case for which the cross-correlation will be identical with the synaptic transmission curve is that in which the firing times of the input axon (a,) are not correlated to the firing times of all the other inputs that affect a 2 and when the firing times of the input obey the statistics of a uniform Poisson process (i.e., the probability of finding a spike is inde- pendent of the time and of the past history of the axon). However, doesn't this depend on the output being a linear function of the input? Well no if we assume that the probability of two spikes being very close together (remember they are very thin in time) is very small, so we can ignore the non-linear effects of two synchronous spikes. (Linear regime)

_{If the input spike train is non-Poissonian, but behaves as a renewal process (i.e., the possibility of finding a spike depends only on the time elapsed since the last spike [Cox, 1962]), then the true synaptic transmission curve can be obtained by deconvolving the apparent re- sponse curve from the autocorrelation curve of the input train [Perkel et al., 1967].}

It is possible to evaluate the amount of correlation introduced by the stimulus itself by computing the so-called shift-predictor correlogram [Perkel et al., 1967; Dickson and Gerstein, 1974]. "joint poststimulus time histogram" (JPSTH). How does this work? => paper – Crosscorrelogram )_{(very clear explanation)}. crosscorrelograms give some measure of the firing rate or firing probability of the target neuron around the time that the reference neuron fires.

Stimulating the cells simultaneously creates a peak in the cross-correlation. This covariation in firing rates of the two stimulated cells must be removed before considering the peak to be relevant (coming from a synaptic connectivity between them or common input). The easiest way to "correct" for this stimulus-induced relationship is to use the shift predictor. See further explanation here

Transmission of firing rate signals through neuronal circuits

Transmission through a chain of synapses.

The effect of a single cell on a population of cells.

The effect of a population of cells on one cell

Transmission of a spike train through a Synapse, computed using Convolution.

Within the linear regime (achieved when the instantaneous firing rate is low enough), one can apply many tools from linear Signal processing and analysis (Linear response theory..) to analyze the input-output relations of complicated neuronal circuits.

Neuronal chains

For successful transmission of Information, we need converging/diverging chains of connected neurons (aka braids).

To summarize, we have seen that in the cortex, activity can be trans- mitted only between sets of cells with multiple diverging/converging connections. We have shown that such connectivity exists and have de- scribed some of the constraints on the size of the sets and the multiplicity of the connections among them. We have suggested that a chain of such connections can operate repeatedly if the activity along such a chain of connections is transmitted by synchronous volleys of spikes (Synfire chain)

Again, In Section 6.3 we saw that transmission through a diverging/converging chain is not stable. It either decays to zero or amplifies itself until it reaches saturation. Our contention, therefore, was that transmission through such a chain was most likely to occur by synchronous volleys. This form of transmission is examined in detail in Chapter 7.

Synchronous transmission

Synchronous gain

Definition 7.1.1: Synchronous gain 50 (SG50). The ratio between the amplitude (A) of an EPSP (see here) and the distance from threshold to the median membrane potential is called the "synchronous gain 50" (SG50). If the membrane potential fluctuates along a symmetric probability density function (e.g., Gaussian p.d.f.), the median is equal to the aver- age, and we have

$SG50 = A/T$

where A is the amplitude of the EPSP, and T is the distance from threshold to the average membrane potential.

We use the median because if the amplitude of the EPSP is equal to distance of the threshold above the median membrane potential, there's a 50% that the membrane potential + EPSP will be above threshold, creating a spike.

In general, if the SG50 of a synapse is x, it takes 1/x such synapses to produce a synchronous response with a probability of 0.5. There's reasons to believe that most neural chains are synfire chains in the cortex, so that they work mostly by seizing synchronous effects.