Foundations of Neuronal Representations - Brain-Machine Interface Engineering - page 43

where p (angle) is the probabilistic density of all the direction angles, which can be easily estimated
by the Parzen window [ 70 ]. The probability of a spike, p (spike), can be simply obtained as the
percentage of the spikes occurring during a large segment of the spike train such that the estimate
is reliable, and the value is not very small. p (spike | angle) is the conditional probability density of
the spike given the direction angle.
For each neuron, the histogram of the spike-triggered angle can be drawn and normalized to
approximate p (spike = 1, angle). In other words, only when there is a spike, the direction angle is
counted for the histogram during the corresponding direction angle bin. Then p (spike = 1| angle)
is approximated by dividing the histogram of (spike = 1, angle) by the histogram of (angle), which
corresponds actually to the Bayesian formulation.

p (spike = 1,angle)
p (angle)

p ( spi ke = 1 | angle ) =

(2.3)

and p (spike = 0|angle) = 1 - p (spike = 1|angle) .

In this way, the mutual information between the kinematic direction angle and the neural
spike train can be estimated for each neuron. Figure 2.12 shows the mutual informa-
tion tuning calculated from three different kinematic vectors for 185 neurons. The M1
(primary motor cortex) tuning clearly shows that the velocity (the middle plot) conveys
more tuning information than position or acceleration, which has been suggested in the
literature, but it is impossible to evaluate in Figure 2.11 because the conventional method
normalizes to the same value all the responses. Therefore, because of the self-normal-
ized definition, the information theoretical tuning metric found that particular neurons
are more tuned to the position, whereas other neurons are more tuned to the velocity.
Because the mutual information is a nonlinear measure, experience has shown that the
information theoretic tuning depth should be plotted in logarithmic scale for BMIs.

2.7.6 Timing Codes
Finding appropriate methodologies to analyze in time the spatial neuronal organization during
motor control, auditory/visual processing, and cognition is a challenging question. Methods for
identifying the fine timing relationships among co-activated neurons have focused on statistical
relationships among observed single spike trains [ 71-76 ] (see Reference [ 77 ] for a review). For pairs
of neuronal recordings, several techniques have been proposed such as cross-intensity functions [ 78 ]
or the method of moments [ 79 ], joint peristimulus time histograms [ 80 ], and methods based on
maximum likelihood (ML) estimation [ 81 ].