Signal Processing by Multiplexing and 
Demultiplexing in Neurons 
David C. Tam 
Division of Neuroscience 
Baylor College of Medicine 
Houston, TX 77030 
dtamCnext-cns.neusc.bcm.tmc.edu 
Abstract 
Signal processing capabilities of biological neurons are 
investigated. Temporally coded signals in neurons can be 
multiplexed to increase the transmission capacity. 
Multiplexing of signal is suggested in bi-threshold neurons with 
"high-threshold" and "low-threshold" for switching firing 
modes. To extract the signal embedded in the interspike- 
intervals of firing, the encoded signal are demultiplexed and 
multiplexed by a network of neurons with delayed-line 
circuitry for signal processing. The temporally coded input 
signal is transformed spatially by mapping the firing intervals 
topographically to the output of the network, thus decoding 
the specific firing interspike-intervals. The network also 
provides a band-pass filtering capability where the 
variability of the timing of the original signal can be decoded. 
1 INTRODUCTION 
Signals of biological neurons are encoded in the firing patterns of spike trains or 
the time series of action potentials generated by neurons. The signal content of 
the codes encoded by a presynaptic neuron will be decoded by some other neurons 
postsynpatically. Neurons are often thought to be encoding a single type of 
282 
Signal Processing by Multiplexing and Demultiplexing in Neurons 283 
codes. But there is evidence suggesting that neurons may encode more than one 
type of signals. One of the mechanisms for embedding multiple types of signals 
processed by a neuron is multiplexing. When the signals are multiplexed, they 
also need to be demultiplexed to extract the useful information transmitted by 
the neurons. Theoretical and experimental evidence of such multiplexing and 
demultiplexing scheme for signal processing by neurons will be given below. 
2 MULIPLEXING IN NEURONS 
Most neurons fire action potentials when the membrane potential is 
depolarized to a threshold above the resting potential. For some neurons, there 
are more than a single threshold that can trigger the generation of action 
potentials. The thresholds occur not only at depolarized membrane potential 
(above the resting potential) but also at hyperpolarized potential (below the 
resting potential). This bi-threshold phenomena had been reported in a number 
of biological neurons including the giant squid axon (Hodgkin & Huxley, 1952), 
thalamic (Jahnsen & Llinfis, 1984), inferior olivary (Yarom & Llinfis, 1987), 
and hippocampal neurons (Stasheff & Wilson, 1990). The phenomena of 
triggering the firing of action potentials at a membrane potential below the 
resting potential level following prolonged hyperpolarization have been 
observed under different conditions in different neurons such as during the 
anodal break after voltage-clamped at a hyperpolarized potential (Hodgkin 
& Huxley, 1952), and are called "low-threshold spikes" (Yarom & Llins, 
1987) and "baseline spikes" (Stasheff & Wilson, 1990), which are spikes 
elicited naturally during the after-hyperpolarization (a.h.p.) period. The 
generation of low-threshold spikes is a voltage- and time-dependent process 
occurring during a prolonged hyperpolarization for de-inactivation of ionic 
conductances. 
Given this bi-threshold for firing of action potentials, a neuron can function in 
two modes of operations: one at depolarization potentials and the other at 
hyperpolarization potentials. Thus, when the neuron is depolarized from the 
resting potential, the neuron will process signal based on the "high-threshold", 
and when the neurons is hyperpolarized for a prolonged duration, the neuron 
will process signal based on the "low-threshold". Formally, it is described as 
follows: 
y(t)= 
O, 
if V(t)  Oh i or 
if V(t-iAt) < Oio and V(t)  Oio, for 1 <i<j 
otherwise 
(1) 
where y(t) denotes the occurrence of the firing of an action potential at time t, 
x(t) denotes the membrane potential of the neuron at time t, Ohi denotes the 
"high-threshold" and 01o denotes the "low-threshold", and jzit represents the 
duration of hyperpolarization, such that the neuron will fire when 
284 Tam 
depolarized at the hyperpolarization potential. This bi-threshold firing 
phenomenon was suggested to be involved in the two different rhythms 
generated by a neuron as a periodic bi-stable oscillator (Rose & Hindmarsh, 
1985; Goldbeter & Moran, 1988), which can switch between two different firing 
frequencies, thus multiplexing the signal depending on the mode of operation or 
polarization level (Tam, 1990c). 
3 DEMULTIPLEXING IN NEURONS 
The multiplexed signal encoded in a neuron can be demultiplexed in a number of 
ways. One of the systematic way of extracting the firing frequency of the 
encoded signal can be described by a network of neurons. Given the temporally 
modulated input spike train spike, the firing intervals of the encoded signal 
can be extracted by a network of neurons such that the firing of these output 
neurons will decode the interspike-intervals of the input signal. In this 
network, the temporal codes of the input spike train will be converted into a 
spatially-distributed topographical code where each output neuron represents 
a particular firing interval with a specific band-width. Thus, the original 
signal is demultiplexed by mapping the input firing intervals into the firing of 
specific neurons based on the spatial location of the neuron in the output layer. 
The circuitry of this network of neurons utilizes delay-lines for signal 
processing (Reiss, 1964; Tam, 1990a, b). Examples of delay-line architecture 
used for signal processing can be found in the cerebellar cortex (Eccles et al., 
1967), inferior colliculus (Yin, et al., 1987, 1986, 1985; Chan et al., 1987) and 
cochlear nucleus (Carr & Konishi, 1990). 
The time-delayed network can be described as follows. Let x(t) be a time-series 
of spikes (or delta-functions, 6(t)) with a total of n+l spikes: 
x(t) =  6(t- 
(2) 
Let the input to the network be a spike train x(t) given by (2). There are k 
neurons in the first input layer of the network. The input is split into multiple 
branches, each of which is connected to all k neurons in the first layer. In 
addition to the direct connection between the input and the first layer neurons, 
each input branch to the first layer neuron is also split into multiple branches 
with successive incremental time-delays. Specially, the k-th neuron in the 
first layer has k+l input lines, each input is successively delayed by a time 
delay At relative to the previous one. That is, the i-th input to this k-th 
neuron in the first layer at time t is given by x(t-iAt). Thus, the sum of the input 
to this k-th neuron is given by: 
Signal Processing by Multiplexing and Demultiplexing in Neurons 285 
k 
Xt)= x(t- iat) 
i=O 
(3) 
3.1 BAND-PASS FILTERING 
Band-pass filtering can be accomplished by the processing at the first layer of 
neurons. If the threshold for the generation of an output spike for the k-th 
neuron is set at one, then this neuron will fire only when the interspike- 
interval Ij, of the input spike train is within the time-delay window, kzit. 
That is, the output of this k-th neuron is given by: 
yg, t)=l, if Xlc> 1 (4) 
O, otherwise 
The interspike-interval, Ij, is defined as the time interval between any two 
adjacent spikes: 
Ij= r i- rj.1, for 0 < jn (5) 
Therefore, the k-th neuron can be considered as encoding a band-pass filtered 
input interspike-intervaL 0 < Ij  kzit. Thus, the k-th neuron in the first layer 
essentially capture the input interspike-interval firing of less than kzit, the 
band-passed interspike-interval. To ensure that the neuron will fire a spike of 
At in duration, we introduce a refractory period of (k-1)zit after the firing of a 
spike for the k-th neuron to suppress continual activation of the neuron due to 
the phase differences of the incoming delayed signal. 
3.2 HIGHER-ORDER INTERSPIKE-INTERVAL PROCESSING 
Higher-order interspike-intervals can be eliminated by the second layer 
neurons. The order of the interspike-interval is defined by the number of 
intervening spikes between any two spikes in the spike train. That is, the first- 
order interspike-interval contains no intervening spike between the two 
adjacent spikes under consideration. Second-order interspike-interval is the 
time interval between two consecutive first-order interspike-intervals, i.e., the 
interval containing one intervening spike. 
If the second layer neurons receive excitatory input from the corresponding 
neuron with a threshold (0 > 1) and inhibitory input from the corresponding 
neuron with a threshold of (0 > 2), then the higher-order intervals are 
eliminated, with the output of the second layer (double-primed) neuron given 
by: 
y,(t)=yg, t).y(t)=l, if2Xg't)> I (6) 
O, otherwise 
where 
286 Tam 
y(t)=l, ifXlc>2 
O, otherwise 
(7) 
This requires that an addition input layer of neurons be added to the network, 
which we call the first-parallel layer, whose input/output relationship is 
given by (7). In other words, there are k first layer neurons and k first-parallel 
layer neurons serving as the input layers of the network. The k-th neuron in the 
first layer and the k-th neuron in the first-parallel layer are similar in their 
inputs, but the thresholds for producing an output spike are different. The 
difference between the outputs of the first set of neurons (first layer) in the first 
layer and the primed set of neurons (first-parallel layer) is computed by the 
second layer by making excitatory connection from the first layer neuron and 
inhibitory connection from the first-parallel layer neuron for each 
corresponding k-th neuron respectively as described by (6). This will ensure 
accurate estimation of only first-order interspike-interval, 0 < Ij  kzit, within 
the time-delay window kzit. 
3.3 BAND-WIDTH PROCESSING 
The third layer neurons will filter the input signal by distributing the 
frequency (or interval) of firing of neurons within a specific band-width. Since 
the k-th neuron in the second layer detects the band-passed first-order 
interspike-intervals (0 < Ij  kzit) and the h-th neuron detects another band- 
passed interspike-intervals (0 < Ij  hat), then the difference between these 
two neurons will detect first-order interspike-intervals with a band-width of 
(k-h)zit. In order words, it will detect the first-order interspike-interval 
between kzit and hat, i.e., hat < Ij  kzit. 
This requires that the third layer neurons derive theix inputs from two sources: 
one excitatory and the other inhibitory from the second layer. The output of 
the k-th neuron in the thixd layer, y"'k(t), is obtained from the difference 
between the outputs of k-th and h-th neurons in the second layer: 
y lch( t) = y'( t) - y h(t) = 
k 
if2 x(t- iAt) > 1 
i=h 
otherwise 
(7) 
A two-dimensional topographical map of the band-passed interspike-intervals 
of the input spike train can be represented by arranging the third-layer neurons 
in a two-dimensional array, with one axis (the horizontal axis) representing 
the k index (the band-passed interspike-interval) of equation (7) and the other 
axis (the vertical axis) representing the (k-h) index (the band-width 
Signal Processing by Multiplexing and Demultiplexing in Neurons 287 
interspike-interval). Thus the firing of the third layer neurons represents the 
band-passed filtered version of the original input spike train, extracting the 
firing interspike-interval of the input signal. The "coordinate" of the neuron in 
the third layer represents the band-passed interspike-interval (0 < Ij kAt) 
and the band-width interspike-interval (hAt < Ij skAt) of the original input 
spike train signal. The band-width can be used to detect the variations (or 
jittering) in the timing for firing of spikes in the input spike train, since the 
timing of firing of spikes in biological neurons can be very variable. Thus, the 
network can be used to detect the variability of timing in firing of spikes by the 
firing location of the third layer neuron. 
3.4 EXTRACTION OF EMBEDDED SIGNAL BY BI-THRESHOLD FIRING 
If the neurons in the second and third layers are bi-threshold neurons where one 
threshold is at the "depolarization" level (i.e., a positive value) and the 
other threshold is at the "hyperpolarization" level (i.e., a negative value), 
then addition information may be extracted based on the level of firing 
threshold. Since the neuron in the second and third layers receive inhibitory 
inputs from the preceding layer, there are instances where the neuron be 
"hyperpolarized" or the sum of the inputs to the neuron is negative. Such 
condition occurs when the order of the interspike-interval is higher than one. 
In other words, the higher-order interspike-interval signal is embedded in the 
"hyperpolarization", which is normally suppressed from generating a spike 
when there is only one threshold for firing at the "depolarized" level (O:i). 
But for bi-threshold neurons where there is another threshold at the 
hyperpolarized level (01o), such embedded signal encoded as 
hyperpolarization can be extracted by sending an external depolarizing signal 
to this neuron causing the neuron to fire at the low threshold. Thus the 
hyperpolarization signal can be "read-out" by an external input to the bi- 
threshold neuron. In summary, a time-delay network can be used to process 
temporally modulated pulsed-coded spike train signal and extract the firing 
interspike-intervals by mapping the band-passed intervals topographically on 
a two-dimensional output array from which the order of the interspike- 
interval can be extracted using different thresholds of firing. 
Acknowledgements 
This work is supported by ONR contract N00014-90-J-1353. 
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