{"title": "Signal Processing by Multiplexing and Demultiplexing in Neurons", "book": "Advances in Neural Information Processing Systems", "page_first": 282, "page_last": 288, "abstract": null, "full_text": "Signal Processing by Multiplexing and \n\nDemultiplexing in Neurons \n\nDavidC. Tam \n\nDivision of Neuroscience \nBaylor College of Medicine \n\nHouston, TX 77030 \n\ndtam@next-cns.neusc.bcm.tmc.edu \n\nAbstract \n\nto \n\nthe \n\nincrease \n\nSignal  processing  capabilities  of  biological  neurons  are \ninvestigated.  Temporally  coded  signals  in  neurons  can  be \nmultiplexed \ntransmission  capacity. \nMultiplexing of signal is suggested in bi-threshold neurons with \nII high-threshold II  and  II low-thre shold II  for  switching  firing \nmodes.  To  extract  the  signal  embedded  in  the  interspike(cid:173)\nintervals  of firing,  the  encoded  signal are  de multiplexed  and \nmultiplexed  by  a  network  of  neurons  with  delayed-line \ncircuitry  for  signal  processing.  The  temporally  coded  input \nsignal is transformed spatially by mapping the firing intervals \ntopographically  to  the  output of  the  network,  thus  decoding \nthe  specific  firing  inters pike-intervals.  The  network  also \nprovides  a  band-pass  filtering  capability  where \nthe \nvariability of the timing of the original signal can be decoded. \n\n1 INTRODUCTION \nSignals of biological neurons are encoded in the firing patterns of spike trains or \nthe time series of action potentials generated by neurons.  The signal content of \nthe codes encoded by a presynaptic neuron will be decoded by some other neurons \npostsynpatically.  Neurons are  often thought to be  encoding a  single  type  of \n\n282 \n\n\fSignal Processing by Multiplexing and Demultiplexing in Neurons \n\n283 \n\ncodes.  But there is evidence suggesting that neurons may encode more than one \ntype of signals.  One of the mechanisms for embedding multiple types of signals \nprocessed by a neuron is multiplexing.  When the signals are multiplexed, they \nalso need to be demultiplexed to extract the useful information transmitted by \nthe neurons.  Theoretical and experimental evidence of such multiplexing and \ndemultiplexing scheme for signal processing by neurons will be given below. \n\n2 MULIPLEXING IN NEURONS \nMost  neurons  fire  action  potentials  when  the  membrane  potential  is \ndepolarized to a threshold above the resting potential.  For some neurons, there \nare  more  than  a  single  threshold  that  can  trigger  the  generation  of  action \npotentials.  The thresholds occur not only at depolarized membrane  potential \n(above  the  resting potential)  but also  at  hyperpolarized  potential  (below  the \nresting potential).  This bi-threshold phenomena had been reported in a number \nof biological neurons including the giant squid axon (Hodgkin & Huxley, 1952), \nthalamic  (Jahnsen  &  Llinas,  1984),  inferior  olivary  (Yarom  &  Llinas,  1987), \nand  hippocampal  neurons  (Stasheff  &  Wilson,  1990).  The  phenomena  of \ntriggering the firing of  action  potentials  at  a  membrane  potential below  the \nresting  potential  level  following  prolonged  hyperpolarization  have  been \nobserved under  different  conditions  in  different  neurons  such  as  during the \nanodal break  after  voltage-clamped  at  a  hyperpolarized  potential  (Hodgkin \n&  Huxley,  1952),  and  are  called  \"low-threshold  spikes\"  (Yarom  &  Llinas, \n1987)  and  \"baseline  spikes\"  (Stasheff  &  Wilson,  1990),  which  are  spikes \nelicited  naturally  during  the  after-hyperpolarization  (a.h. p.)  period.  The \ngeneration  of low-threshold spikes  is  a  voltage- and time-dependent  process \noccurring during a  prolonged  hyperpolarization  for  de-inactivation  of ionic \nconductances. \nGiven this bi-threshold for  firing of action potentials,  a neuron can function  in \ntwo  modes  of  operations:  one  at  depolarization  potentials  and  the  other  at \nhyperpolarization potentials.  Thus,  when the neuron is  depolarized from  the \nresting potential, the neuron will process signal based on the  \"high-threshold\", \nand when the neurons is hyperpolarized for  a  prolonged duration, the neuron \nwill  process signal based  on the  \"low-threshold\".  Formally,  it  is  described  as \nfollows: \n\nI,  ifV(t)~Ohi \n\nor \n\n{\nyet) = \n\nif V(t-iilt) < OlD  and vet) ~ OlD'  for 1 <i<j \n\n(1) \n\n0,  otherwise \n\nwhere yet)  denotes the  occurrence of the firing of an action potential at time  t, \nx(t)  denotes  the  membrane  potential of the  neuron  at time  t,  0hi  denotes  the \n\"high-threshold\"  and  OlD  denotes  the  \"low-threshold\",  and jilt  represents  the \nduration  of  hyperpolarization,  such  that  the  neuron  will  fire  when \n\n\f284 \n\nTam \n\ndepolarized  at  the  hyperpolarization  potential.  This  bi-threshold  firing \nphenomenon  was  suggested  to  be  involved  in  the  two  different  rhythms \ngenerated by  a  neuron  as  a  periodic bi-stable  oscillator  (Rose  & Hindmarsh, \n1985; Goldbeter & Moran, 1988), which can switch between two different firing \nfrequencies, thus multiplexing the signal depending on the mode of operation or \npolarization level (Tam,  1990c). \n\n3 DEMULTIPLEXING IN NEURONS \nThe multiplexed signal encoded in a neuron can be demultiplexed in a number of \nways.  One  of  the  systematic  way  of extracting  the  firing  frequency  of  the \nencoded signal can be described by a network of neurons.  Given the temporally \nmodulated  input spike  train  spike,  the  firing intervals  of  the  encoded signal \ncan be extracted by a  network of neurons such that the firing of these  output \nneurons  will  decode  the  interspike-intervals  of  the  input  signal. \nIn  this \nnetwork,  the  temporal codes  of the input spike  train will be converted into a \nspatially-distributed topographical code where each output neuron represents \na  particular  firing  interval  with  a  specific  band-width.  Thus,  the  original \nsignal is demultiplexed by mapping the input firing intervals into the firing of \nspecific neurons based on the spatial location of the neuron in the output layer. \nThe  circuitry  of  this  network  of  neurons  utilizes  delay-lines  for  signal \nprocessing  (Reiss,  1964;  Tam,  1990a,  b).  Examples  of delay-line  architecture \nused for  signal processing can be  found  in the cerebellar  cortex (Eccles  et  al., \n1967),  inferior  colliculus  (Yin,  et al.,  1987,  1986,  1985;  Chan et  al.,  1987)  and \ncochlear nucleus (Carr & Konishi, 1990). \nThe time-delayed network can be described as follows.  Let x(t) be a time-series \nof spikes (or delta-functions,  6(t\u00bb)  with a total of n+l spikes: \n\nn \n\nx(t) = ~ 6(t- T) \n\nj=O \n\n(2) \n\nLet  the  input  to  the  network be  a  spike  train  x(t)  given  by  (2).  There  are  k \nneurons in the first input layer of the network.  The input is split into multiple \nbranches,  each  of  which is  connected  to  all k  neurons in the  first  layer.  In \naddition to the direct connection between the input and the first layer neurons, \neach input branch to the first layer neuron is  also split into multiple branches \nwith  successive  incremental  time-delays.  Specially,  the  k-th  neuron  in  the \nfirst  layer  has  k+ 1  input lines,  each  input is  successively  delayed  by  a  time \ndelay  Lit  relative  to  the  previous  one.  That  is,  the  i-th  input  to  this  k-th \nneuron in the first layer at time t is given by x(t-iLit).  Thus, the sum of the input \nto this k-th neuron is given by: \n\n\fSignal Processing by Multiplexing and Demultiplexing in Neurons \n\n285 \n\nk \n\nXit) = :Lx(t- it1t) \n\ni=O \n\n(3) \n\n3.1 BAND-PASS FILTERING \nBand-pass filtering can be accomplished by the  processing at  the first  layer of \nneurons.  If the  threshold for  the  generation  of an  output spike  for  the  k-th \nneuron  is  set  at  one,  then  this  neuron  will  fire  only  when  the  inters pike(cid:173)\ninterval,  Ij,  of  the  input  spike  train  is  within  the  time-delay  window,  kAt. \nThat is, the output of this k-th neuron is given by: \n{ I,  ifXk>l \n0,  otherwise \n\nt) = \n\n(4) \n\ny.,/. \n\nThe  interspike-interval, Ij ,  is  defined  as  the  time  interval  between  any  two \nadjacent spikes: \n\n(5) \nTherefore, the k-th neuron can be  considered as encoding a band-pass filtered \ninput  interspike-interval,  0 < Ij  $  kAt.  Thus, the  k-th  neuron in the  first layer \nessentially  capture  the  input  interspike-interval  firing  of  less  than  kAt,  the \nband-passed interspike-interval  To ensure that the neuron will fire  a spike of \nAt  in duration, we introduce a refractory  period of (k-I)At  after the firing of a \nspike for  the k-th neuron to suppress continual activation of the neuron due to \nthe phase differences of the incoming delayed signal. \n\n3.2 HIGHER-ORDER INTERSPIKE-INTERVAL PROCESSING \nHigher-order  inters pike-intervals  can  be  eliminated  by  the  second  layer \nneurons.  The  order  of  the  interspike-interval is  defined  by  the  number  of \nintervening spikes between any two spikes in the spike train.  That is,  the first(cid:173)\norder  inters pike-interval  contains  no  intervening  spike  between  the  two \nadjacent  spikes  under  consideration.  Second-order inters pike-interval is  the \ntime  interval between  two  consecutive first-order interspike-intervals, i.e.,  the \ninterval containing one intervening spike. \nIf the  second layer  neurons  receive  excitatory  input from  the  corresponding \nneuron with a  threshold (0) 1)  and inhibitory input from  the  corresponding \nneuron  with  a  threshold  of  (0  > 2),  then  the  higher-order  intervals  are \neliminated, with the output of the second layer (double-primed) neuron given \nby: \n\n\" \ny'i/t) =y.,/.t)-y1!t) = \n\n,{I, if2~X.,/.t\u00bb  1 \n\n. \n\n0,  otherwIse \n\n(6) \n\nwhere \n\n\f286 \n\nTam \n\n, \ny1!t)= \n\n{1,  ifXk>2 \n\n0,  otherwise \n\n(7) \n\nThis requires that an addition input layer of neurons be added to the network, \nwhich  we  call  the  first-parallel  layer,  whose  input/ output  relationship  is \ngiven by (7).  In other words, there are k first layer neurons and k first-parallel \nlayer neurons serving as the input layers of the network.  The k-th neuron in the \nfirst  layer  and  the  k-th  neuron  in  the  first-parallel  layer  are  similar  in  their \ninputs, but the  thresholds  for  producing an  output  spike  are  different.  The \ndifference between the outputs of the first set of neurons (first layer) in the first \nlayer  and  the  primed set  of  neurons  (first-parallel  layer)  is  computed by the \nsecond  layer by making excitatory connection from  the first  layer neuron and \ninhibitory  connection  from  the  first-parallel  layer  neuron  for  each \ncorresponding  k-th  neuron respectively as  described by (6).  This  will ensure \naccurate  estimation  of  only first-order  interspike-interval,  0 < Ij  ~ kt1t,  within \nthe time-delay window  kt1t. \n\n3.3 BAND-WIDTH PROCESSING \nThe  third  layer  neurons  will  filter  the  input  signal  by  distributing  the \nfrequency (or interval) of firing of neurons within a specific band-width.  Since \nthe  k-th  neuron  in  the  second  layer  detects  the  band-passed  first-order \ninters pike-intervals  (0  < Ij  ~ kt1t)  and the h-th  neuron  detects  another  band(cid:173)\npassed  interspike-intervals  (0  < Ij  ~ hL1t),  then  the  difference  between  these \ntwo  neurons  will  detect  first-order  interspike-intervals  with  a  band-width of \n(k-h)L1t. \nIn  order  words,  it  will  detect  the  first-order  interspike-interval \nbetween  kL1t  and hL1t,  i.e., hL1t  < Ij  ~ kL1t. \nThis requires that the third layer neurons derive their inputs from two sources: \none excitatory and the other inhibitory  from  the second layer.  The output of \nthe  k-th  neuron  in  the  third  layer,  y\" 'k(t),  is  obtained  from  the  difference \nbetween the outputs of k-th and h-th neurons in the second layer: \n\nY'k'tlt) = y'i/t) - y'h(t) = \n\nk \n\nif2 ~ :Lx(t- iL1t) > 1 \n\ni=h \n\n(7) \n\n,  otherwise \n\nA two-dimensional topographical map of  the band-passed interspike-intervals \nof the input spike train can be represented by arranging the third-layer neurons \nin a  two-dimensional  array,  with one  axis  (the  horizontal  axis)  representing \nthe k index (the band-passed interspike-interval)  of equation (7)  and the other \naxis  (the  vertical  axis)  representing  the  (k-h)  index  (the  band-width \n\n\fSignal Processing by Multiplexing and Demultiplexing in Neurons \n\n287 \n\ninterspike-interval).  Thus the firing  of the third layer neurons represents the \nband-passed  filtered  version  of the  original input spike  train,  extracting the \nfiring interspike-interval of the input signal.  The \"coordinate\" of the neuron in \nthe  third  layer  represents  the  band-passed  interspike-interval  (0 < Ij  :r; kAt) \nand  the  band-width  interspike-interval  (hAt  < Ij \n:r; kAt)  of the  original input \nspike  train  signal.  The band-width  can  be  used  to  detect  the  variations  (or \njittering)  in  the  timing for  firing  of spikes in  the  input spike  train,  since  the \ntiming of firing of spikes in biological neurons can be very variable.  Thus, the \nnetwork can be used to detect the variability of timing in firing of spikes by the \nfiring location of the third layer neuron. \n\n3.4 EXTRACTION OF EMBEDDED SIGNAL BY BI-THRESHOLD FIRING \nIf the neurons in the second and third layers are bi-threshold neurons where one \nthreshold  is  at  the  \"depolarization\"  level  (Le.,  a  positive  value)  and  the \nother  threshold  is  at  the  \"hyperpolarization\"  level  (Le.,  a  negative  value), \nthen  addition  information  may  be  extracted  based  on  the  level  of  firing \nthreshold.  Since  the neuron in the  second and third layers receive inhibitory \ninputs  from  the  preceding  layer,  there  are  instances  where  the  neuron  be \n\"hyperpolarized\"  or  the  sum  of  the  inputs  to  the  neuron  is  negative.  Such \ncondition occurs when the  order of the interspike-interval is  higher than one. \nIn  other words, the higher-order interspike-interval signal is embedded in the \n\"hyperpolarization\",  which  is  normally  suppressed  from  generating  a  spike \nwhen  there  is  only  one  threshold  for  firing  at  the  \"depolarized ll  level  (Ohi)' \nBut  for  bi-threshold  neurons  where  there  is  another  threshold  at  the \nhyperpolarized \nsuch  embedded  signal  encoded  as \nhyperpolarization can be extracted by sending an external depolarizing signal \nto  this  neuron  causing  the  neuron  to  fire  at  the  low  threshold.  Thus  the \nhyperpolarization  signal  can  be  \"read-out\"  by  an  external  input  to  the  bi(cid:173)\nthreshold  neuron.  In  summary,  a  time-delay network can be used  to  process \ntemporally  modulated  pulsed-coded  spike  train  signal  and  extract  the  firing \ninterspike-intervals by mapping the band-passed intervals topographically  on \na  two-dimensional  output  array  from  which  the  order  of  the  interspike(cid:173)\ninterval can be extracted using different thresholds of firing. \n\nlevel \n\n(Olo), \n\nAcknowledgements \nThis work is supported by ONR contract N00014-90-J-1353. \n\nReferences \nCarr,  C.  E.  &  Konishi,  M.  (1990)  A  circuit  for  detection  of  interaural  time \ndifferences in the brain stem of the barn owl.  ].  Neurosci.  10: 3227-3246. \nChan, J.  C., Yin, T.  C.  &  Musicant, A.  D.  (1987)  Effects of interaural time delays \nof noise stimuli on low-frequency cells in the cat1s inferior colliculus.  II. \nResponses to band-pass filtered noises.  ].  Neurophysiol.  58:  543-561. \n\n\f288 \n\nTam \n\nGoldbeter,  A.  &  Moran,  F.  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