{"title": "Multiple Timescales of Adaptation in a Neural Code", "book": "Advances in Neural Information Processing Systems", "page_first": 124, "page_last": 130, "abstract": null, "full_text": "Multiple times cales of adaptation in a neural \n\ncode \n\nAdrienne L. Fairhall, Geoffrey D. Lewen, William Bialek, \n\nand Robert R. de Ruyter van Steveninck \n\nNEe Research Institute \n4 Independence Way \n\nPrinceton, New Jersey 08540 \n\nadrienne!geofflbialeklruyter@ research. nj. nec. com \n\nAbstract \n\nMany neural systems extend their dynamic range by  adaptation.  We ex(cid:173)\namine the  timescales  of adaptation  in  the  context of dynamically  mod(cid:173)\nulated rapidly-varying stimuli,  and demonstrate in the  fly  visual system \nthat  adaptation  to  the  statistical  ensemble  of the  stimulus  dynamically \nmaximizes information transmission about the time-dependent stimulus. \nFurther, while the rate response has  long transients, the adaptation takes \nplace on timescales consistent with optimal variance estimation. \n\n1  Introduction \n\nAdaptation was one of the first phenomena discovered when Adrian recorded the responses \nof single sensory neurons [1,  2].  Since that time, many different forms  of adaptation have \nbeen found in almost all  sensory systems.  The simplest forms of adaptation, such as  light \nand dark adaptation in  the visual system,  seem to  involve just discarding a large constant \nbackground signal so that the system can maintain sensitivity to  small changes.  The idea \nof Attneave [3]  and Barlow [4]  that the nervous system tries to find  an efficient representa(cid:173)\ntion of its sensory inputs implies that neural coding strategies should be adapted not just to \nconstant parameters such as  the mean light intensity, but to  the entire distribution of input \nsignals [5];  more generally, efficient strategies for processing (not just coding) of sensory \nsignals must also be matched to  the statistics  of these signals  [6].  Adaptation to  statistics \nmight happen on evolutionary time scales,  or, at the opposite extreme, it might happen in \nreal time as  an animal moves through the world.  There is  now evidence from several sys(cid:173)\ntems for real time adaptation to statistics [7, 8, 9], and at least in one case it has been shown \nthat the form  of this  adaptation indeed does  serve  to  optimize the efficiency  of represen(cid:173)\ntation, maximizing the information that a single neuron transmits about its  sensory inputs \n[10]. \nPerhaps  the  simplest of statistical  adaptation experiments,  as  in  Ref [7]  and  Fig.  1,  is  to \nswitch between stimuli that are drawn from different probability distributions and ask how \nthe  neuron responds to  the  switch.  When we  'repeat'  the  experiment we  repeat the  time \ndependence of the parameters describing the distribution, but we choose new signals from \nthe same distributions; thus we probe the response or adaptation to the distributions and not \nto  the particular signals.  These switching experiments typically reveal transient responses \nto the switch that have rather long time scales, and it is tempting to identify these long time \nscales as  the  time scales of adaptation.  On  the other hand,  one can also  view  the process \nof adapting  to  a distribution  as  one  of learning  the  parameters  of that distribution,  or of \naccumulating evidence that the  distribution has changed.  Some features  of the  dynamics \n\n\fin  the switching experiments match the dynamics of an  optimal  statistical  estimator [11], \nbut the overall time scale does not:  for all  the experiments we have seen, the apparent time \nscales of adaptation in a switching experiment are much longer than would be required to \nmake reliable estimates of the relevant statistical parameters. \n\nIn this work we re-examine the phenomena of statistical adaptation in the motion sensitive \nneurons of the fly  visual  system.  Specifically, we are  interested in  adaptation to  the vari(cid:173)\nance or dynamic range of the velocity distribution [10].  It has been shown that,  in  steady \nstate,  this adaptation includes a rescaling of the neuron's input/output relation,  so that the \nsystem seems to encode dynamic velocity signals in relative units;  this allows the system, \npresumably, to deal both with the \"-'  50\u00b0 /s motions that occur in straight flight and with the \n\"-'  2000\u00b0 /s motions that occur during acrobatic  flight  (see Ref.[12]).  Further, the precise \nform of rescaling chosen by  the fly's  visual  system is  that which maximizes  information \ntransmission.  There are  several natural questions: (1) How long does it take the system to \naccomplish the rescaling of its input/output relation? (2) Are the transients seen in switch(cid:173)\ning experiments an indication of gradual rescaling? (3) If the system adapts to the variance \nof its inputs, is  the neural signal ambiguous about the absolute scale of velocity?  (4)  Can \nwe see the optimization of information transmission occurring in real time? \n\n2  Stimulus structure and experimental setup \n\nA fly  (Calliphora vicina) is immobilized in wax  and views a computer controlled oscillo(cid:173)\nscope display while we record action potentials from the identified neuron HI using  stan(cid:173)\ndard methods.  The stimulus movie is a random pattern of dark and light vertical bars, and \nthe entire pattern moves along a random trajectory with velocity S(t); since the neuron is \nmotion (and not position) sensitive we refer to this signal as the stimulus. We construct the \nstimulus S(t)  as the product of a normalized white noise s(t), constructed from  a random \nnumber sequence refreshed every Ts  = 2 ms, and an amplitude or standard deviation (J'(t) \nwhich varies on a characteristic timescale Ta  \u00bb Ts.  Frames of the movie are drawn every \n2 ms.  For analysis all spike times are discretized at the 2 ms resolution of the movie. \n\na) \n\nT = 20s \n--- T= lOs \n---T=4s \n\n~ \n\n0 \n\n\"0 \nQ) \n_!!1 \ntil \nE o  -1 \n\nZ \n\n-2+-------~------~------~------~ \n1.00 \n\n0.00 \n\n0.50 \n\n0. 25 \n\n0.75 \n\nNormalised time  1fT \n\n10 \n\n20 \n\n30 \n\n40 \n\nPeriod T  (sec) \n\nFigure 1: (a) The spike rate measured in response to a square-wave modulated white noise stimulus \nset) , averaged over many presentations of set), and normalized by the  mean and standard deviation. \n(b)  Decay time of the rate following an upward switch as a function  of switching period T. \n\n3  Spike rate dynamics \n\nSwitching experiments as described above correspond to a stimulus such that the amplitude \n(J'(t)  is  a square wave,  alternating between two  values  (J'l  and (J'2,  (J'l  > (J'2.  Experiments \nwere  performed over a  range  of switching  periods  (T  =  40,  20,  10, 4 s),  with  the  am(cid:173)\nplitudes  (J'l  and  (J'2  in  a ratio  of 5:1.  Remarkably, the timescales  of the response depend \n\n\fstrongly  on  those  of the  experiment;  in  fact,  the response times  rescale  by T, as  is  seen \nin Fig.  lea).  The decay  of the rate in  the first half of the experiment is fitted  by  an expo(cid:173)\nnential,  and in Fig.  l(b), the resulting decay time T(T)  is  plotted as  a function of T;  we \nuse an exponential not to insist that this is  the correct form, only to extract a timescale.  As \nsuggested by the rescaling of Fig.  lea), the fitted decay times are well described as a linear \nfunction of the stimulus period.  This demonstrates that the timescale of adaptation of the \nrate is not absolute, but is a function of the timescale established in the experiment. \nLarge sudden changes in  stimulus variance might trigger special mechanisms, so  we turn \nto  a signal  that changes  variance continuously:  the  amplitude (J'(t)  is  taken  to  be  the  ex(cid:173)\nponential of a sinusoid, (J'(t)  =  exp(asin(27l'kt)), where the period T  =  11k was  varied \nbetween 2 s and 240 s, and the constant a is fixed such that the amplitude varies by a factor \nof 10 over a cycle. A typical averaged rate response to the exponential-sinusoidal stimulus \nis  shown in Fig.  2(a). The rate is close to  sinusoidal over this parameter regime, indicating \na logarithmic encoding of the  stimulus  variance.  Significantly, the rate response shows  a \nphase lead ~ <I>  with respect to the stimulus.  This may be interpreted as  the effect of adap(cid:173)\ntation:  at every point on the cycle, the gain of the response is  set to  a value defined by the \nstimulus a short time before. \n\n2 \n\na) \n\n:;-\nQl \n-0 \n\n1 \n~ \n.l!l \n0 \n~ \n-0 \nQl \n.!!l \n'iii \nE \n0 \nZ \n\n-2 \n\n-1 \n\n120 \n\n100 \n0- 80 \nQl \n.!E.-\n00  60 \n:c  40 \nrn \nQl \nE  20 \ni= \n\n0 \n\n-20 \n\nT =30 s \n--- T=60s \n- T=90s \n-\n-\n- T= 120s \n\n0.00 \n\n0.25 \n\n0.50 \n\n0.75 \n\n1.00 \n\nNormalised tIT \n\nb) \n\no \n\n100  150  200  250  300 \n\n50 \nPeriod T (sec) \n\nFigure 2:  (a) The spike rate measured in response to a exponential-sinusoidal modulation of a white \nnoise  stimulus  set) , averaged over presentations  of set) , and normalised by  the  mean  and standard \ndeviation,  for  several periods T.  (b)  The time  shift 0 between response  and  stimulus, for a range  of \nperiods T. \n\nAs  before, the response of the system was  measured over a range of periods T.  Fig.  2(b) \nshows the measured relation of the timeshift 8(T) = T ~<I> of the response as  a function of \nT.  One observes that the relation is nearly linear over more than one order of magnitude in \nT; that is, the phase shift is approximately constant. Once again there is a strong and simple \ndependence of the apparent timescale of adaptation on the stimulus parameters. Responses \nto stimulus sequences composed of many frequencies also exhibit a phase shift, consistent \nwith that observed for the single frequency experiments. \n\n4  The dynamic input-output relation \n\nBoth  the  switching  and  sinusoidally  modulated  experiments  indicate  that  responses  to \nchanging the variance of input signals have multiple time scales, ranging from a few  sec(cid:173)\nonds to  several minutes.  Does it really take the system this long to adjust its input/output \nrelation to  the  new  input distribution?  In  the range of velocities used,  and  at the contrast \nlevel  used in  the laboratory,  spiking in  HI  depends on features  of the  velocity waveform \nthat occur within a window of '\"  100 ms.  After a few  seconds, then,  the  system has had \naccess to several tens  of independent samples of the motion  signal,  and should be  able to \nestimate its  variance to  within'\"  20%;  after a minute the  precision  would be better than \na few  percent.  In  practice,  we  are changing the input variance not by  a few  percent but a \n\n\ffactor of two or ten; if the system were really efficient, these changes would be detected and \ncompensated by adaptation on much shorter time scales.  To address this, we look directly \nat the input/output relation as the standard deviation u(t) varies in time. \nFor  simplicity  we  analyze  (as  in  Ref.  [10])  features  of the  stimulus  that  modulate  the \nprobability  of occurrence of individual  spikes,  P(spikelstimulus);  we  will  not consider \npatterns  of spikes,  although  the  same  methods  can  be  easily  generalised.  The  space  of \nstimulus  histories  of length  '\"  100 ms,  discretised  at  2 ms,  leading  up  to  a  s(like  has  a \ndimensionality'\" 50, too large to allow adequate sampling of P(spikelstimulus) from the \ndata, so we must begin by reducing the dimensionality of the stimulus description. \nThe simplest way to  do so is  to  find  a subset of directions in stimulus space determined to \nbe  relevant for the  system,  and  to  project the  stimulus  onto that  set of directions.  These \ndirections correspond to  linear filters.  Such  a set of directions can  be obtained from  the \nmoments  of the  spike-conditional stimulus;  the  first  such  moment is  the  spike-triggered \naverage, or reverse correlation function [2].  It has been shown [10] that for HI, under these \nconditions,  there  are  two  relevant  dimensions:  a  smoothed  version  of the  velocity,  and \nalso  its  derivative.  The rescaling observed in  steady  state  experiments was  seen  to  occur \nindependently in both dimensions, so without loss of generality we will use as our filter the \nsingle dimension given  by  the  spike-triggered average.  The  stimulus  projected  onto this \nfilter will be denoted by So. \nThe filtered stimulus is then passed through a nonlinear decision process akin to a threshold. \nTo calculate the input/output relation P(spikelso) [10], we use Bayes' rule: \n\nP(spikelso) \nP(spike) \n\nP(solspike) \n\nP(so) \n\n(1) \n\nThe  spike  rate  r(so)  is  proportional to  the  probability of spiking,  r(so)  ex:  P(spikelso), \nleading to the relation \n\nr(so) \n\nr \n\nP(solspike) \n= -----',=-:-----:---'--\n\nP(so) \n\n(2) \n\nwhere r is  the mean  spike rate.  P(so)  is  the (lrior distribution  of the projected stimulus, \nwhich  we  know.  The  distribution  P(solspike)  is  estimated  from  the  projected  stimulus \nevaluated at the spike times, and the ratio of the two is  the nonlinear input/output relation. \nA number of experiments have shown that the filter characteristics of HI are adaptive, and \nwe see this in the present experiments as well:  as  the amplitude u(t) is decreased, the filter \nchanges  both in overall  amplitude  and  shape.  The filter  becomes  increasingly extended: \nthe system integrates over longer periods of time under conditions of low  velocities.  Thus \nthe filter depends on the input variance, and we expect that there should be an observable \nrelaxation  of the  filter  to  its  new  steady  state  form  after  a  switch  in  variance.  We  find, \nhowever,  that within 200 ms  following  the  switch,  the amplitude of the  filter has  already \nadjusted to  the  new  variance,  and further that  the detailed  shape  of the  filter has  attained \nits  steady  state  form in  less  than  I  s.  The  precise  timescale  of the  establishment of the \nnew  filter  shape depends on  the value of u:  for the change to  U1,  the steady  state form  is \nachieved within 200 ms.  The long tail of the low variance filter for U2  \u00ab  (1) is established \nmore slowly.  Nonetheless, these time scales which characterize adaptation of the filter are \nmuch shorter than the rate transients  seen in the  switching experiments, and are closer to \nwhat we might expect for an efficient estimator. \nWe construct time dependent input/output relations by forming conditional distributions us(cid:173)\ning spikes from particular time slices in a periodic experiment.  In Figs.  3.I(b) and 3.I(c), \nwe show  the  input/output relation calculated in  1 s bins  throughout the switching experi(cid:173)\nment.  Within the first  second the input/output relation is almost indistinguishable from its \nsteady state form.  Further, it takes the same form for the two halves of the experiment: it is \nrescaled by  the standard deviation, as was  seen for the steady state experiments. The close \ncollapse or rescaling of the input/output relations depends not only on the normalisation by \nthe standard deviation, but also on the use of the \"local\" adapted filter (i.e.  measured in  the \nsame time  bin).  Returning to  the  sinusoidal  experiments, the  input/output relations  were \n\n\f1b) \n\n2 \n\n.  1a) \n\n\";. -\n\n-1 \n\n10  , - - - - - - . - - - - - - - - ,  r----.,,--------,--,------, r - - - - - - - - - - - - - ,  \n\n10 \n\n15 \n\n20 \n\n10 \n\n15 \n\n20 \n\n25 \n\n30 \n\n20 \n\n40 \n\n60 \n\n80 \n\n\" > \n~('J \n\n001 \n\n10 \n\n3a) \n\n\" \n..... ili \n-.:: \n\n0 1 \n\n0.01 \n\nFigure 3: Input/output relations for (a) switching, (b) sinusoidal and (c) randomly modulated exper(cid:173)\niments.  Figs.  3.1  show the modulation envelope u(t) , in log for (b)  and (c) (solid), and the measured \nrate  (dotted),  normalised  by  mean  and  standard  deviation.  Figs.  3.2  show  input/output  relations \ncalculated in  non-overlapping bins  throughout  the  stimulus  cycle, with  the  input  80  in  units of the \nstandard  deviation  of the  whole  stimulus.  Figs.  3.3  show  the  input/output relations  with  the  input \nrescaled to units of the local standard deviation. \n\nconstructed for T  = 45 s in 20 non-overlapping bins of width 2.25 s.  Once again the func(cid:173)\ntions  show a remarkable rescaling which is  sharpened by  the use of the appropriate local \nfilter:  see Fig.3.2(b) and (c).  Finally, we consider an amplitude which varies randomly with \ncorrelation time Tu  '\" 3 s:  u(t) is a repeated segment of the exponential of a Gaussian ran(cid:173)\ndom process, pictured in Fig.3.3(a), with periodicity T  = 90s\u00bb Tu.  Dividing the stimulus \ninto sequential bins  of 2 s in width, we obtain the  filters  for each timeslice,  and calculate \nthe local prior distributions, which are not Gaussian in this case as they are distorted by the \nlocal variations of u(t).  Nonetheless, the ratio P(solspike)j P(so) conspires such that the \nform of the input/output relation is preserved. \nIn  all  three  cases,  our results  show  that  the  system  rapidly  and  continuously  adjusts  its \ncoding strategy, rescaling the input/output relation with respect to the local  variance of the \ninput as for steady state stimuli. Variance normalisation occurs as rapidly as is measurable, \nand the system chooses a similar form for the input/output relation in each case. \n\n5 \n\nInformation transmission \n\nWhat does this mean for the coding efficiency of the neuron? An experiment was designed \nto  track the information transmission  as  a  function  of time.  We  use a  small  set of N  2 s \nlong random noise  sequences  {Si(t)},  i  = 1, ... ,N, presented in  random order at two \n\n\fdifferent amplitudes, 0\"1  and 0\"2.  We then ask how much information the spike train conveys \nabout  (a)  which  of the  random  segments  Si(t)  and  (b)  which  of the  amplitudes  O\"j  was \nused.  Specifically, the experiment consists of a series of trials  of length 2 s where the fast \ncomponent is  one of the sequences {Si}, and after 1 s,  the amplitude switches from  0\"1  to \n0\"2  or vice versa.  N  was taken to be 40, so that a 2 hour experiment provides approximately \n80 samples for each  (Si,O\"j).  This  allows  us  to measure the  mutual information between \nthe response and either the fast or the slow component of the stimulus as a function of time \nacross  the  2  s repeated segment.  We  use  only  a very  restricted  subspace of 0\"  and s:  the \nmaximum available information about 0\"  is  1 bit, and about sis log2N. \nThe spike response is represented by \"words\" [13], generated from  the spike times discre(cid:173)\ntised  to  timebins  of 2 ms,  where no spike is  represented by 0,  and a spike by  1.  A word \nis  defined as  the  binary digit formed  from  10 consecutive bins,  so  there  are  210  possible \nwords.  The information about the fast component S  in trials of a given 0\"  is \n\nIcr(w(t); s) = H[Pcr(w(t))]  - L P(sj)H[Pcr(w(t); Sj)]' \n\nN \n\nj=l \n\nwhere H  is the entropy of the word distribution: \n\nH[P(w(t))]  = - L P(Wk(t)) log2 P(Wk(t)). \n\nk \n\n(3) \n\n(4) \n\nOne can  compare this  information for different values of 0\".  Similarly,  one can calculate \nthe information about the amplitude using a given probe s: \n\nIs (w(t); 0\")  = H[Ps(w(t))]- L P(O\"j)H[Ps(w(t); O\"j)]. \n\n2 \n\n(5) \n\nj=l \n\nThe amount of information for each S j  varies rapidly depending on the presence or absence \nof spikes, so we average these contributions over the {Sj} to give I(w; 0\"). \n\n__ I(w;s).  upward sWitch \n\no \n\nI(w;s).  downward SWitch \n\n-/>- I(w;,,) \n\n-075 \n\n-050 \n\n-0.25 \n\n000 \n\n025 \n\n0.50 \n\n075 \n\nTime relative to switch (sec) \n\nFigure 4:  Information per spike as a function of time where 0\"  is switched every 2 s. \n\nThe mutual information as  a function of time is  shown in Fig.  4,  presented as  bits/spike. \nAs one would expect, the amount of information transmitted per second about the stimulus \ndetails, or s, depends on  the ensemble parameter 0\":  larger velocities allow a higher SNR \nfor  velocity  estimation,  and  the  system  is  able  to  transmit  more  information.  However, \n\n\fwhen we convert the information rate to  bits/spike, we find  that the system is  transmitting \nat a constant efficiency of around 1.4 bits/spike.  Any  change in information rate during a \nswitch from  0\"1  to  0\"2  is  undetectable.  For a switch from  0\"2  to  0\"1,  the time to recovery is \nof order 100 ms.  This demonstrates explicitly that the system is indeed rapidly maximising \nits information transmission.  Further, the transient \"excess\" of spikes following an upward \nswitch provide information at a constant rate per spike.  The information about the ampli(cid:173)\ntude, similarly, remains at a constant level throughout the trial.  Thus, information about the \nensemble variable is retained at all times:  the response is not ambiguous with respect to the \nabsolute scale of velocity.  Despite the rescaling  of input/output curves, responses within \ndifferent ensembles are distinguishable. \n\n6  Discussion \n\nWe  find  that  the  neural  response  to  a  stimulus  with  well-separated  timescales  S(t)  = \nO\"(t)s(t)  takes  the form of a ratel8)timing code, where the response r(t)  may be  approxi(cid:173)\nmately modelled as \n\nr(t)  =  R[O\"(t)]g (s(t)). \n\n(6) \nHere R modulates the overall rate and depends on the slow dynamics of the variance enve(cid:173)\nlope,  while the precise timing of a given spike in response to  fast events in the stimulus is \ndetermined by the nonlinear input/output relation g, which depends only on the normalised \nquantity s(t). Through this apparent normalisation by the local standard deviation, g, as for \nsteady-state experiments, maximises information transmission  about the fast components \nof the  stimulus.  The function R modulating the rate varies on much slower timescales  so \ncannot be taken as  an indicator of the extent of the system's adaptation to a new ensemble. \nRather, R appears to function as an independent degree of freedom, capable of transmitting \ninformation, at a slower rate,  about the slow  stimulus modulations.  The presence of many \ntimescales in  R may  itself be an  adaptation to  the  many timescales of variation in natural \nsignals.  At  the  same  time,  the  rapid  readjustment of the  input/output relation  - and  the \nconsequent recovery of information after a sudden change in 0\"  -\nindicate that the adaptive \nmechanisms approach the limiting speed set by the need to gather statistics. \n\nAcknowledgments \n\nWe thank B.  Agiiera y Arcas, N. Brenner and T.  Adelman for helpful discussions. \n\nReferences \n\n[1]  E. Adrian (1928) The Basis of Sensation (London Christophers) \n[2]  F.  Rieke, D. Warland, R.  de Ruyter van Steveninck and W. 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Strong, R.  Koberle, R. de  Ruyter van  Steveninck and W.  Bialek (1998)  Phys. Rev.  Lett.  80, \n\n197-200. \n\n\f", "award": [], "sourceid": 1809, "authors": [{"given_name": "Adrienne", "family_name": "Fairhall", "institution": null}, {"given_name": "Geoffrey", "family_name": "Lewen", "institution": null}, {"given_name": "William", "family_name": "Bialek", "institution": null}, {"given_name": "Robert", "family_name": "van Steveninck", "institution": null}]}