Using the original input of 34 spikes in A as a reference,
the average frequency was then decreased by increments of
1 spike by randomly removing the corresponding number of
spikes to reach the desired average frequency. Ten input
patterns were tested per frequency level. For each of these
patterns the maximal response was simulated and an average
calculated per frequency level. The results are shown in Fig 5B dark triangles.

The frequency was then increased from 35 to 70 spikes by
randomly adding the desired number of spikes to the input
pattern. Ten input patterns were tested for each frequency
level. For each of these patterns the maximal response was
simulated and an average calculated per frequency level. The
results are shown in Fig 5B open triangles. We see that
when the input frequency is double (100%) that of the
original, an increase in response of only 7.2% is observed
compared to the original response.

To obtain the input represented in C, a temporal noise
filter of ±3ms was applied to the original spatio-temporal
pattern shown in A. The total number of spikes remained
constant as well as the number of spikes emitted from each
RF neuron; therefore the average frequency remained
unchanged as before. D depicts the response for this input:
the membrane activity remained relatively weak throughout
most of the input duration then increased sharply to a
maximum of 94.3mV above the resting potential, 28% less
than the response to the original spike pattern. The response
of this temporal neuron decreased as the temporal noise
increased on the input: Fig 5A triangle signs.

Using the spike pattern represented in B and applying a
random noise filter of maximum ±3ms gave the
configuration represented by D. This filter involved the
random modification of the firing times within a maximum
range of 3ms before or after the original spike timing. The
total number of spikes remained constant and hence also the
average firing rate. Fig 3E shows the response for this
particular input. The membrane potential increased gradually
until reaching a maximum of about 115mV above the resting
potential.

I developed a C++ software to simulate the dendritic computation implemented by equations 1 to 5. This software simulates a single neuron with a single dendrite. This neuron receives inputs from a neuronal layer called receptive field (RF) neurons all of which are spiking point neurons. Using this software, we calculated the membrane potential at each position on the dendrite and at the soma for differing numbers of synapses, synaptic positions and spike input patterns.

The configuration tested first was that of a frequential neuron with an 80μm long dendrite, a passive propagation velocity: v=1 μm.ms-1, and one RF neuron as input projecting 20 regularly spaced synapses. Each synapse had a synaptic weight of 6 mV, a rise time constant of 1ms and a general time constant of 2ms.

A first spike train of 33 spikes was used to stimulate the dendrite computing (DC) neuron: B. C shows the response for this particular input. The membrane potential increased gradually until reaching a maximum at about 115mV above the resting potential that was arbitrarily set to 0mV.

11 RF neurons were used as the input, all projecting to the
same DC neuron with a total of 42 synapses on its dendrite.
The precise morphology of this network has been elaborated
according to the rules explained in section II.D and Fig 2. A
depicts the original input activity. Each line of the raster plot
represents the temporal activity of one of the 11 RF neurons.
The bottom-most line represents the sum of the 11 lines
above. A depicts the response of the DC neuron for this
input: the membrane activity remained relatively weak
during most of the input duration then increased sharply to a
maximum of 130.9 mV above the resting potential.

A temporal neuron like in C can detect specific spatio-temporal spike
patterns. On the top, an example of a spatio-temporal spike pattern based on three neurons N1 to N3 and 6 spikes. On the bottom, the specific organization of synapses along the dendrite corresponding to the detected spike pattern given above.

A neuron for which the behavior can be described by eq. 4
(hereafter named temporal neuron) is able to detect a
specific spatio-temporal pattern of spikes. The general idea
of this form of detection is that all the EPSPs created by the
different spikes in the spike train will converge at the same
moment at the soma using the different propagation times to
counter-balance the different spike arrival times at each
synapse; all this creating a maximal depolarization in the
soma. A different version of the same general idea was
suggested by Gerstner et al. ([7] page 144) with the notable
difference that instead of using different propagation times,
they implemented different rise times.

Certain studies have showed that synapses have a stable
influence that is independent of their distance to the soma, a
phenomenon referred to as “equal vote” or “dendritic
democracy” [12], [13]. In this setting, the synaptic weights
and time constant would be modified in order to obtain the
same EPSP at the soma level for all the synapses i.e.
distance-dependent scaling [14]. If we consider that the
propagation times remain different, we can calculate a
composite EPSP with stable amplitude of several tens of
milliseconds. To obtain this type of effect in the soma it is
necessary to organize the synapses in a regular manner along
the dendrite with the synaptic weights increasing with
distance to the soma. We will call this particular
configuration the “frequency sensitive setting” and all the
others “spatio-temporal sensitive”. Here we demonstrate that
neurons organized in a frequency sensitive manner (hereafter
named frequential neurons) are able to « count » the spikes
that arrive within a specific time interval, a phenomenon
which itself depends on the dendritic length and the
propagation speed. With a constant EPSP shape, the neuron
becomes particularly sensitive to the average firing rate of
the global pattern but reacts little to changes in the precise
timing of each spike

Parallel algorithms for Artificial Inteligence (AI)