to characterize coding and adaptation in a large population of cortical neurons , we recorded spiking activity from the primary visual cortex ( v1 ) of anesthetized cats using 10 10 electrode arrays ( fig .
we characterized responses as a function of stimulus orientation using sequences of static gratings with random orientation and phase , each presented for 32 ms ( fig .
we considered stimulus ensembles with two statistical distributions : uniform and biased . in the uniform case , the probability of each orientation was equal .
in the biased case , the probability of one orientation was markedly higher than the others .
as shown previously , population responses to stimuli with uniform statistics could be accurately fitted on the basis of the tuning curves of the neurons .
1e ) and the population responses varied as a function of time and of the preferred orientation of the neurons ( fig .
we divided the axis of preferred orientation evenly into bins , each pooling the activity of neurons with similar orientation preferences .
, we obtained tuning curves for each of the bins of preferred orientation ( fig .
these tuning curves are homogeneous : they are similar to each other except for their preferred orientation .
as expected , the tuning curves could be used to fit the population responses to the stimulus sequence through summation ( fig . 1 g ) . this simple operation ( followed by a mild nonlinearity ) captured a high proportion ( 63% ) of the explainable variance in the population responses in this experiment .
when we changed the statistics of the stimulus ensemble , the cortex displayed a remarkable ability to adapt ( fig .
we biased the stimulus sequence markedly in favor of one orientation ( the  adaptor  ) , presenting it three times more often than the rest ( fig .
1i ) . based on the tuning curves obtained with the uniform stimulus ensemble ( fig .
1c ) the neurons selective for the adaptor orientation should respond on average much more than the others ( fig . 1k ) .
the actual response of the population , instead , showed no such bias : the average over time of the cortical response to the biased stimulus ensemble did not show significant variation across neurons differing in preferred orientation ( fig .
1j ) . from the cortical responses it was not apparent that one orientation was shown three times more than the others .
evidently the neurons in visual cortex had adapted to the biased ensemble , and their adaptation had been strong enough to completely counteract the bias in the ensemble . moreover ,
this adaptation was not so strong as to overcompensate , i.e. as to create a  hole  in the population responses .
the effects of adaptation were well described by a new set of tuning curves , one that was tailored for the biased ensemble ( fig .
1d ) by applying regression on the population responses to the biased ensemble ( fig .
this fit captured 40% of the explainable variance in the responses , significantly higher than the 22% captured by the homogeneous tuning curves .
in particular , the adapted tuning curves correctly captured how the average population response would show no trace of the bias in the stimulus ( fig .
the adapted tuning curves were significantly different from those measured during the presentation of the uniform ensemble , both in this experiment ( fig .
1c , d ) and the others ( suppl . figs . 2 and 3 ) . indeed ,
if one used the adapted tuning curves to simulate the responses to the uniform ensemble , one would grossly underestimate the responses of some of the neurons ( fig . 1h , and suppl .
2 ) . to pool the results across sessions we gave the orientation of the adaptor the nominal value of 0 deg , and the average response to the uniform ensemble the nominal value of 1 ( fig .
2d ) , showing neither a peak nor a trough for the responses of neurons tuned to the adaptor .
looking at these average responses , it would be impossible to tell that the adaptor had been shown 3 - 5 times more than the other orientations .
indeed , there was no statistical difference between the average responses of neurons selective for the adaptor ( 0 deg ) and for the orthogonal orientation ( t - test , p=0.98 ) . as we have seen in the example experiment ( fig .
1 ) , therefore , adaptation changed the tuning curves just as needed to counter the bias in the stimulus ensemble  neither too much nor too little . just as we have seen in the example data set ( fig .
no single set of tuning curves could predict the measured responses to both the uniform and biased stimulus ensembles .
2d ) could be fitted by the adapted tuning curves ( capturing 54%20% of the explainable variance , se , n = 4 , fig .
2f ) , but could not be obtained from the homogeneous tuning curves , which predicted a non - existing peak in the population responses at the orientation of the adaptor ( 16%12% of the explainable variance , fig .
conversely , the responses measured with the uniform stimulus ensemble could be fitted by the homogeneous tuning curves ( 88%16% of the explainable variance , n = 8 , fig .
2b ) , but could not be obtained with the adapted tuning curves , which predicted a non - existing hole in the population responses at the orientation of the adaptor ( capturing 48%11% of the explainable variance , fig . 2c ) .
this analysis , therefore , reveals that adaptation maintains equality in the time - average activity of different neurons , in the face of biases in the stimulus ensemble .
achieving this equality requires appropriate calibration : if adaptation were stronger or weaker there would be a valley or a mountain in the population responses , and instead there is neither .
equality , however , can only be maintained for a range of stimulus biases . at the extreme ,
if a single orientation were shown 100% of the time ( as is typically the case in previous studies of adaptation ) one would expect the neurons selective for that orientation to respond more than the rest .
indeed , while we saw complete equalization when the probability of the adaptor was 30 - 40% ( fig .
2e ) , equalization was less perfect when we increased this probability to 50% ( suppl .
because of the width of the tuning curves , a biased stimulus sequence tends to engage not only the neurons selective for the adaptor orientation but also those selective for nearby orientations . in the absence of adaptation
, the activity of these neurons would therefore become highly correlated ( a  correlation catastrophe  ) . to look for these effects , we computed the correlation coefficients between pairs of bins , for each combination of orientation preference .
these are known as  signal  correlations , and they reflect the similarity in tuning curves . as expected , the matrix of correlations in the uniform case was diagonal ( fig .
3b ) , but they also predict that in the responses to the biased stimulus ensemble there should be a strong central peak in the matrix ( fig .
3d ) showed a diagonal structure of correlations that is similar to the one seen with the uniform stimulus ensemble ( fig .
consistent with a longstanding theoretical proposal , therefore , adaptation prevented responses of cortical neurons from becoming more correlated .
this effect of decorrelation could be captured by the adapted tuning curves : running the biased stimulus ensemble through these adapted curves resulted in a diagonal matrix of correlations without a central peak ( fig .
the adapted tuning curves , instead , would not have been appropriate in response to the uniform stimulus ensemble , as they would have caused a large central hole in the matrix of correlations ( fig .
these results indicate that adaptation provides two homeostatic effects to the population , maintaining equality not only in the first order statistics , but also in the second order statistics .
these effects are achieved quickly , with an average time constant of 1.7  0.4 s ( s.d .
in such a short interval the cortex was able to engage adaptation mechanisms that effectively counteracted the bias in the stimulus ensemble , both in terms of first and of second order statistics . however , adaptation is known to operate on more than one time scale .
perhaps a relevant determinant of time scale is the number of stimuli that adaptation mechanisms need to observe to be fully engaged . in our experiments ,
it was sufficient for v1 to observe the same orientation in 16 - 26 of those stimuli to drastically adapt its responsiveness and selectivity .
what does adaptation change in a population to allow it to discount these stimulus statistics ?
since the effects of adaptation are captured by changes in tuning curves , the answer lies in the attributes of these tuning curves ( fig .
to characterize these tuning curves , and to reveal more detailed effects , we examined the full matrix of responses to individual flashing gratings , averaging the results of all our experiments ( fig .
4b ) depends both on the preferred orientation of the neurons ( rows ) and on the orientation of the stimulus ( columns ) .
taking sections across the rows yields the familiar tuning curves of each preference bin ( fig .
4a ) . taking sections across columns instead yields population response profiles , one for each stimulus orientation ( fig .
the response matrix measured with the biased ensemble summarizes the effects of adaptation ( fig .
4d - f ) . a qualitative look at these tuning curves and population responses indicates that adaptation had two main effects .
the first effect was an offset : a small reduction of all responses , regardless of orientation preference ( fig .
the second effect was a marked reduction in amplitude of the tuning curves , which as strongest for the tuning curves of neurons with orientation preference near zero , the nominal orientation of the adaptor ( fig .
this effect amounted to creating a hole in the diagonal of the matrix ( fig .
there are two simple ways to creating such a hole along the diagonal of the response matrix ( fig .
the first possibility is intuitive and rests on neuron identity : adaptation would reduce mostly the responses of the neurons selective for the adaptor ( fig .
the second possibility is less intuitive , and rests on the stimuli rather than on the neurons : adaptation would control the population responsiveness to different stimuli , reducing it most strongly for stimuli with the same orientation as the adaptor ( fig .
the neuron - specific description of adaptation evokes simple forms of neural fatigue , and agrees with the view of adaptation as controlling a neuron 's responsiveness or sensitivity .
the stimulus - specific description involves fatiguing stimuli ( rather than neurons ) , reducing their effectiveness in driving the cortex .
this description agrees with the stimulus - specific effects of adaptation that have been reported in single neurons . to resolve the dichotomy in these descriptions , we designed a simple model that includes both neuron - specific and stimulus - specific factors ( fig .
4g - i ) . in the model , the response matrix is scaled by two multiplicative gain factors .
one of these factors specifies how much to reduce the responsiveness of each neuron ( fig .
4i ) and the other specifies how much to reduce the responses to each stimulus ( fig
we described the gain factors as gaussians peaking at zero ( the orientation of the adaptor and the preferred orientation of neurons tuned for the adaptor ) .
, the model works as follows : take the response matrix measured with the uniform stimulus ensemble ( fig .
the model is defined by only five parameters : the two widths and two amplitudes of the gaussians , and the constant offset .
this similarity is confirmed by plotting the tuning curves on top of the measured ones ( fig .
4j ) or by plotting the population response profiles on top of the measured ones ( fig .
similar results were obtained when working with matrices obtained separately from the experiments with adaptor probability of 30 - 40% ( n=6 , 80.0% of the variance ) and with adaptor probability of 50% ( n=5 , 86.7% of the variance ) . in particular , the model correctly predicted that adaptation would reduce the tuning curves and repel them away from the adaptor orientation ( fig .
5a - c ) . because one of the gain factors depends on stimulus orientation ( fig .
4 g ) , it scales the tuning curves more on the flank towards the adaptor than on the other flank , pushing them away from the adaptor orientation ( fig .
in addition , the model makes a novel prediction : that adaptation should not only reduce population response profiles but also repel them away from the adaptor ( fig .
5p - r ) . the second gain factor depends on preferred orientation ( fig .
4i ) , so it scales the population profiles more on the flank where neurons are selective for the adaptor than on the opposite flank , pushing them away from the adaptor orientation .
5e ) and the peak of the population response profile is pushed away from the adaptor orientation ( fig .
because it accounts for responses simultaneously recorded from a population , the model allows us to measure the relative importance of stimulus - specific adaptation and neuron - specific adaptation .
remarkably , the less intuitive component of the model , the one that is stimulus - specific ( fig .
the reduction in stimulus - specific gain was consistently larger than the reduction in neuron - specific gain , both in individual experimental sessions and in averages across sessions ( suppl .
. the most important effect of adaptation is therefore the one that is stimulus - specific , as if adaptation had reduced the effective strength of stimuli with orientation near the adaptor .
our analysis of the effects of adaptation at the level of populations has concerned the average responses that the neurons give to repeated stimulus presentations ( commonly termed  signal  ) , and not the trial - by - trial deviations from these average responses (  noise  ) . measurements performed in awake primates following traditional , prolonged adaptation to single stimuli suggest that adaptation reduces the correlation among these deviations (  noise correlations  ,
28 ) in neuronal pairs . under our experimental conditions , however , we found little evidence for such an effect ( fig .
indeed , adaptation to a biased ensemble reduced noise correlations in some pairs , but increased it in others ( fig .
6b ) : in some , adaptation slightly reduced noise correlations ( 6/11 with p<0.05 : wilcoxon rank sum test ) , and in others it slightly increased them ( 4/11 ) , or showed no significant difference ( 1/11 ) . on average , however , noise correlations stayed quite constant : the average difference in noise correlation between experiments with biased statistics and experiments with uniform statistics was a negligible 0.020.06 ( s.d . ,
similar effects were seen when the spike binning was shifted relative to the stimulus refresh times ( not shown ) , and when noise correlations were computed over longer time intervals ( 2 s bins rather than 32 ms bins , suppl . fig .
we then asked whether adaptation might influence noise correlations more for pairs that are selective for certain orientations than for others .
the results were negative , with similarly low effects of adaptation on noise correlations regardless of the preferred orientation in a pair , relative to the adaptor ( suppl .
these results indicate that under our experimental conditions the effects of adaptation on the population code can be fully summarized by the effects on average signals and signal correlations , with little effect on trial - by - trial deviations from these signals .
to summarize , we discovered that primary visual cortex displays a remarkable ability to counteract biases in the stimulus ensemble , by rapidly introducing the appropriate opposing biases in the responsiveness and selectivity of neurons .
these adaptation phenomena are due to homeostatic mechanisms that have two simple goals : to maintain equality in the time - averaged responses across the population , and to enforce independence in selectivity across the population .
the first goal , equalization , is consistent with proposals made by psychophysicists of adaptation being a  graphic equalizer  to counteract changes in the statistics of the environment .
it also echoes proposals that adaptation may act to  centre  a population response by subtracting the responses to the prevailing stimulus distribution , or to scale responses so that the average of a measured signals is always mapped onto a fixed internal representation .
the second goal , independence , was predicted by a longstanding proposal concerning the role of adaptation in maintaining decorrelation in cortex .
we were able to understand the goals of these homeostatic mechanisms because we measured responses in a whole population , and because we measured activity concurrently with changes in stimulus statistics .
these stimuli allowed us to observe the homeostatic mechanisms achieve their stable ( and arguably intended ) effects rather than their fleeting ( and arguably unintended ) aftereffects .
by contrast , previous studies of adaptation in v1 used the traditional adapt - test technique developed in psychophysics , where the phases of adaptation and response measurement are distinct .
this technique can only reveal adaptation 's aftereffects , those that persist after a change in stimulus properties even though they are no longer needed .
we further discovered that adaptation in primary visual cortex follows a very simple arithmetical rule to shape the population responses . at the heart of this rule
is multiplication with two gain factors , one that depends on stimulus attributes and one that depends on neuronal preference .
this rule provides a unified framework that encompasses the known effects of adaptation on responsiveness and orientation selectivity of individual v1 neurons . having this arithmetical rule , in turn , can guide and constrain research into the underlying circuits and mechanisms .
for instance , since our recordings mostly targeted layer 2/3 , the adaptation we measured there could be at least partially inherited from inputs from layer 4 .
the mechanisms we have discovered may well operate in the whole cortex , but for now we can only demonstrate that they do so in primary visual cortex , and specifically in the processing of stimulus orientation .
future work could establish the degree to which our observations generalize to other stimulus variables beyond orientation , and to other cortical areas beyond v1 .
further work is also required to understand how the effects of adaptation in one sensory area cascade into subsequent ones , leading to compounding perceptual effects .
for instance , the effects of adaptation observed in primary visual cortex appear to be quite distinct from those observed in cortical area mt ( refs .
it is not entirely clear to what degree these differences can be explained by inheritance .
moreover , though it is possible that the effects of adaptation in area mt would be consistent with those that we propose , this needs to be established experimentally , for instance by adapting mt neurons to aspects of the stimuli that do not elicit selective responses in v1 .
we tend to notice its perceptual effects only when it does not work properly , or rather when it is catching up with a marked change in stimulus statistics .
the simple arithmetical rules that we have uncovered to describe neural adaptation may guide future work to study this perceptual adaptation .
moreover , they provide a framework to link adaptation with other mechanisms of homeostasis , such as the ones at work during plasticity and development .
meanwhile , the results we have observed in primary visual cortex offer a promising first glimpse at how an entire cortical population adapts to the statistics of its inputs .
experiments were carried out at the smith - kettlewell eye research institute , under the supervision of the local institutional animal care and use committee .
they were performed in the same animals and with the same methods as in our previous study .
briefly , 4 young adult female cats ( 2 - 4 kg ) were anesthetized with ketamine and xylazine during surgical procedures and with sodium pentothal and fentanyl during electrical recordings .
the animals were respirated and the depth of anesthesia carefully monitored and adjusted by following eeg and vital signs .
utah probes ( blackrock , utah ) were inserted in area v1 with a pneumatic device to minimize tissue damage , and covered in 2% agar to improve stability .
the probes consisted of a grid of 1010 silicon electrodes with 400 m spacing and 1.5 mm electrode length .
insertion depths were about 0.81 mm , resulting in recordings confined mostly to layers 2/3 .
receptive fields were typically on the vertical meridian , indicating that the electrodes were placed at the border between areas 17 and 18 , which together form cat area v1 .
stimuli were presented to the contralateral eye on a crt monitor ( refresh rate 120 hz , mean luminance 32 cd / m ) .
they consisted of large , stationary gratings ( 30 deg in diameter ) flashed in random sequence for 32 ms each .
each grating had one of 4 spatial phases and one of 6 - 12 orientations .
the orientations could occur either with equal probability or with a biased statistics , where the  adaptor orientation had a higher probability of occurrence ( p = 0.30 , 0.35 , 0.40 , or 0.50 ) .
grating contrast was typically 50 - 80% and spatial frequency was 0.2 cycles / deg .
this spatial frequency was empirically determined to be effective in activating the recorded neurons , and is visible to neurons in both area 17 and 18 ( ref . ) .
segments were presented in random order and each block of segments was generally presented ten times .
traces were acquired at 12 khz and firing rates were obtained by low - pass filtering the spike trains with a cutoff at 25 hz .
firing rates were then resampled at 32 ms intervals ( the duration of our flashed gratings ) . to identify multi - unit activity ( mua ) we set thresholds at 4 s.d . of the background noise in each of the 96 channels of the multielectrode array .
population responses were computed by binning sites ( 15 deg bin width ) according to their preferred orientation .
such preferred orientation was determined in the first experiment of the series with an unbiased stimulus sequence using event - related analysis . binned responses in subsequent experiments with a biased stimulus ensemble where normalized to the time averages of responses to the first experiment . to compute correlation coefficients , instead ,
to study the effects of adaptation we could not use the traditional method of event - related analysis because this method works only for random stimuli whose distribution is uniform and spherical .
we therefore measured a neuron 's filter ( or receptive field ) f( , )in time  and orientation  as the least - square solution to the equation r(t ) = s(t   , )f( ,  ) , where r(t ) is the response of the neuron at time t and s(t   ,  ) is the stimulus at time t   and orientation  ( a matrix of zeros and ones ) .
we solved the equation in matlab version r2012a using the pseudoinverse operator pinv . in the case of uniform stimuli , this procedure is analogous to simple stimulus - triggered averaging ( suppl .
other aspects of the ln model , including the static nonlinearity , are as in our previous study .
the static nonlinearity was imposed to be the same when modeling the responses to uniform and biased stimuli ; this ended up being an excellent approximation as the fits barely improved if we allowed two different nonlinearities in the two adaptation conditions .
only experiments with average responses r(t ) having a mean over variance ( s / n ) significantly lower at times before the stimulus onset than at times during peak response ( ~40 ms after stimulus onset ) were considered for further analysis . the filters f(
, )were matrices of size 12  8 , where 12 was the number of preferred orientations and 8 the number of time steps in the past ( suppl . fig .
we obtained one - dimensional tuning curves by considering the time at which the filter attained its maximal value . to study how adaptation affected the correlations between stimuli ( fig .
then , we normalized the data by the standard deviation over time of the responses in the uniform - ensemble condition . finally , we computed the covariance between pairs of orientation bins : cov = e[(ri  e(ri))(rj  e(rj ) ) ] , where e is the expectation operator , and ri(t ) and rj(t ) are the responses of neurons in bins i and j. the model of adaptation ( fig 4g - l ) specifies the responses to individual gratings of neurons selective for orientation p to stimulus with orientation s .
the response of the neurons when the stimulus ensemble is uniform is runiform(s , p ) , and the responses when the stimulus ensemble is biased is rbiased(s , p ) . the model can be written as follows rbiased(s,p)=s(sa)p(pa)runiform(s,p)k , where a is the adaptor orientation . in this equation
,  is an overall gain factor , k is a subtractive term , and s and p are terms that govern the stimulus - specific adaptation and the neuron - specific adaptation : s(sa)=1asg(sa,28.3),p(pa)=1apg(pa,28.3 ) , where g( ,  ) is a circular gaussian with mean  and standard deviation . the value of 28.3 deg was simply obtained by fitting a gaussian function to the tuning curves . in other words : runiform(s,p)g(sp,28.3 ) .
the subtractive baseline k can be taken to be a constant or , for improved fits , to be a function of stimulus orientation as follows : k(sa)=kg(sa,k ) . here ,
the parameters k and k determine the strength and tuning of the subtractive term . in the fits ,
the latter came out to be very large , leading to an almost flat curve , i.e. almost a constant .
the model , therefore , has five free parameters : the three multiplicative gain factors  , as , and ap , and the two parameters of the subtractive baseline , k and k . to compute noise correlations , we measured multiunit activity on 44 - 96 responsive recording sites per experiment .
sites were considered responsive if their mean response explained at least 10% of the variance across repeats .
we divided time in 32 ms or 2 s bins and represented the spike count at time t of the i site to the n repeat of a stimulus sequence as ri , n(t ) .
the noise correlations between a pair of sites i and j are calculated as the pearson 's linear correlation coefficient between ( ri , n(t ) 
rj(t ) ) , where rk ( t ) is the mean response of site k across all stimulus repeats .
we also considered the time bins 1.5 s after stimulus onset , to allow for the effects of adaptation to stabilize .
 sensory systems exhibit mechanisms of neural adaptation , which adjust neuronal activity based on recent stimulus history . in primary visual cortex ( v1 ) 
 , in particular , adaptation controls the responsiveness of individual neurons and shifts their visual selectivity . what benefits does adaptation confer to a neuronal population ? 
 we measured adaptation in the responses of populations of cat v1 neurons to stimulus ensembles with markedly different statistics of stimulus orientation . 
 we found that adaptation serves two homeostatic goals . 
 first , it maintains equality in the time - averaged responses across the population . 
 second , it maintains independence in selectivity across the population . 
 adaptation scales and distorts population activity according to a simple multiplicative rule that depends on neuronal orientation preference and on stimulus orientation . 
 we conclude that adaptation in v1 acts as a mechanism of homeostasis , enforcing a tendency towards equality and independence in neural activity across the population . 
Results
Discussion
Methods
Supplementary Material