Assuming a random distribution of patterns, the exact probability of a false match is given by:
! !!! ×
(1)
The denominator is simply the total number of possible patterns containing A active cells in a population of N total cells. The numerator counts the number of patterns that would connect to θ or more of the s synapses on one dendritic segment. A more detailed description of this equation can be found in (Ahmad and Hawkins, 2015).
The equation shows that a non-linear dendritic segment can robustly classify a pattern by sub-sampling (forming synapses to only a small number of the cells in the pattern to be classified). Table A in S1 Text lists representative error probabilities calculated from Eq.(1).
By forming more synapses than necessary to generate an NMDA spike, recognition becomes robust to noise and variation. For example, if a dendrite has an NMDA spike threshold of 10, but forms 20 synapses to the pattern it wants to recognize, twice as many as needed, it allows the dendrite to recognize the target pattern even if 50% of the cells are changed or inactive. The extra synapses also increase the likelihood of a false positive error. Although the chance of error has increased, Eq.(1) shows that it is still tiny when the patterns are sparse. In the above example, doubling the number of synapses and hence introducing a 50% noise tolerance, increases the chance of error to only 1.6 x 10^-18 . Table 1B in S1 Text lists representative error rates when the number of synapses exceeds the threshold.
