Discovering flexible codes for prediction across timescales in the retina

Discovering flexible codes for prediction across timescales in the retina | bioRxiv /* */ /* */ <!-- <!-- /*! * yepnope1.5.4 * (c) WTFPL, GPLv2 */ (function(a,b,c){function d(a){return"[object Function]"==o.call(a)}function e(a){return"string"==typeof a}function f(){}function g(a){return!a||"loaded"==a||"complete"==a||"uninitialized"==a}function h(){var a=p.shift();q=1,a?a.t?m(function(){("c"==a.t?B.injectCss:B.injectJs)(a.s,0,a.a,a.x,a.e,1)},0):(a(),h()):q=0}function i(a,c,d,e,f,i,j){function k(b){if(!o&&g(l.readyState)&&(u.r=o=1,!q&&h(),l.onload=l.onreadystatechange=null,b)){"img"!=a&&m(function(){t.removeChild(l)},50);for(var d in y[c])y[c].hasOwnProperty(d)&&y[c][d].onload()}}var j=j||B.errorTimeout,l=b.createElement(a),o=0,r=0,u={t:d,s:c,e:f,a:i,x:j};1===y[c]&&(r=1,y[c]=[]),"object"==a?l.data=c:(l.src=c,l.type=a),l.width=l.height="0",l.onerror=l.onload=l.onreadystatechange=function(){k.call(this,r)},p.splice(e,0,u),"img"!=a&&(r||2===y[c]?(t.insertBefore(l,s?null:n),m(k,j)):y[c].push(l))}function j(a,b,c,d,f){return q=0,b=b||"j",e(a)?i("c"==b?v:u,a,b,this.i++,c,d,f):(p.splice(this.i++,0,a),1==p.length&&h()),this}function k(){var a=B;return a.loader={load:j,i:0},a}var l=b.documentElement,m=a.setTimeout,n=b.getElementsByTagName("script")[0],o={}.toString,p=[],q=0,r="MozAppearance"in l.style,s=r&&!!b.createRange().compareNode,t=s?l:n.parentNode,l=a.opera&&"[object Opera]"==o.call(a.opera),l=!!b.attachEvent&&!l,u=r?"object":l?"script":"img",v=l?"script":u,w=Array.isArray||function(a){return"[object Array]"==o.call(a)},x=[],y={},z={timeout:function(a,b){return b.length&&(a.timeout=b[0]),a}},A,B;B=function(a){function b(a){var a=a.split("!"),b=x.length,c=a.pop(),d=a.length,c={url:c,origUrl:c,prefixes:a},e,f,g;for(f=0;f<d;f++)g=a[f].split("="),(e=z[g.shift()])&&(c=e(c,g));for(f=0;f<b;f++)c=x[f](c);return c}function g(a,e,f,g,h){var i=b(a),j=i.autoCallback;i.url.split(".").pop().split("?").shift(),i.bypass||(e&&(e=d(e)?e:e[a]||e[g]||e[a.split("/").pop().split("?")[0]]),i.instead?i.instead(a,e,f,g,h):(y[i.url]?i.noexec=!0:y[i.url]=1,f.load(i.url,i.forceCSS||!i.forceJS&&"css"==i.url.split(".").pop().split("?").shift()?"c":c,i.noexec,i.attrs,i.timeout),(d(e)||d(j))&&f.load(function(){k(),e&&e(i.origUrl,h,g),j&&j(i.origUrl,h,g),y[i.url]=2})))}function h(a,b){function c(a,c){if(a){if(e(a))c||(j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}),g(a,j,b,0,h);else if(Object(a)===a)for(n in m=function(){var b=0,c;for(c in a)a.hasOwnProperty(c)&&b++;return b}(),a)a.hasOwnProperty(n)&&(!c&&!--m&&(d(j)?j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}:j[n]=function(a){return function(){var b=[].slice.call(arguments);a&&a.apply(this,b),l()}}(k[n])),g(a[n],j,b,n,h))}else!c&&l()}var h=!!a.test,i=a.load||a.both,j=a.callback||f,k=j,l=a.complete||f,m,n;c(h?a.yep:a.nope,!!i),i&&c(i)}var i,j,l=this.yepnope.loader;if(e(a))g(a,0,l,0);else if(w(a))for(i=0;i (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0];var j=d.createElement(s);var dl=l!='dataLayer'?'&l='+l:'';j.src='//www.googletagmanager.com/gtm.js?id='+i+dl;j.type='text/javascript';j.async=true;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-M677548'); Skip to main content Home About Submit ALERTS / RSS Search for this keyword Advanced Search New Results Discovering flexible codes for prediction across timescales in the retina Kyle Bojanek , Baptiste Lefebvre , Jared Salisbury , Olivier Marre , Stephanie E. Palmer doi: https://doi.org/10.1101/2025.09.19.677348 Kyle Bojanek 1 Department of Organismal Biology and Anatomy, University of Chicago , Chicago, Illinois 60637, USA Find this author on Google Scholar Find this author on PubMed Search for this author on this site Baptiste Lefebvre 2 Institut de la Vision, Sorbonne Université, INSERM, CNRS , Paris, France Find this author on Google Scholar Find this author on PubMed Search for this author on this site Jared Salisbury 1 Department of Organismal Biology and Anatomy, University of Chicago , Chicago, Illinois 60637, USA Find this author on Google Scholar Find this author on PubMed Search for this author on this site Olivier Marre 2 Institut de la Vision, Sorbonne Université, INSERM, CNRS , Paris, France Find this author on Google Scholar Find this author on PubMed Search for this author on this site Stephanie E. Palmer 1 Department of Organismal Biology and Anatomy, University of Chicago , Chicago, Illinois 60637, USA 3 Department of Physics, University of Chicago , Chicago, Illinois 60637, USA 4 Physics Frontier Center for Living Systems, University of Chicago , Chicago, Illinois 60637, USA Find this author on Google Scholar Find this author on PubMed Search for this author on this site For correspondence: sepalmer{at}uchicago.edu Abstract Full Text Info/History Metrics Preview PDF Abstract Efficient coding theory postulates that a sensory system maximizes information between its response and the input, yet it is unclear if a different measure of optimality that takes into account output function might give a better fit to neural data. The sensory processing delays in many systems suggest that the maximization of predictive information is a reasonable objective function for driving fast, effective downstream behavior. We introduce a one-parameter family of optimal encoding distributions based on how far out in time a population of retinal ganglion cells is optimized to predict future stimuli. Analyzing the population response to a moving bar stimulus with rich temporal correlation structure identifies which particular optimal encoding best describes the neural activity. This allows for the discovery of how far out in time the retina is predicting, instead of simply testing for optimality at one timescale. As stimulus statistics change, so too does the time scale of prediction that best matches the population response. Focusing on this optimal timescale, the neural code can be evaluated in terms of classic efficient coding theory, revealing that the code also shows a peak in how these predictive bits are allocated in the population response repertoire. The stimulus has a fully controlled set of temporal statistics, but is still complex enough to show behaviors like starts and stops, constant motion, and motion reversals. Its tractable statistical structure allows for an information theoretic account of computations like motion anticipation and the retina reversal response in terms of the maximization of predictive information. INTRODUCTION Efficient coding, the theory that a population of sensory neurons should maximize mutual information between the response R and the stimulus S subject to constraints on neural firing rates ( Fig. 1 ), is one of the most influential normative theories in sensory processing [ 1 , 2 ]. In linear systems, information is determined by the correlation between stimulus and response, but for more complex nonlinear stimuli and responses, mutual information provides a general measure [ 3 ]. Mutual information, I ( R ; S ) = H ( R ) − H ( R | S ), quantifies how much uncertainty about the response R is reduced by knowing the stimulus S . Maximizing I ( R ; S ) can be achieved either by increasing response entropy (diversity), H ( R ), or reducing noise, H ( R | S ). Maximizing mutual information includes many strategies based on both maximizing population entropy H ( R ) (including stimulus decorrelation [ 2 , 4 , 5 ] and high entropy response to input features [ 6 ]) and minimizing conditional entropy H ( R | S ) by decreasing response noise [ 7 , 8 ]. Information maximization has effectively explained aspects of retinal ganglion cell responses, including center-surround receptive field structure [ 9 ], contrast sensitivity functions [ 4 ], retinal mosaic geometry [ 10 – 13 ], response nonlinearities [ 14 ], response statistics [ 15 ], and cell-type diversity [ 16 ]. Download figure Open in new tab FIG. 1. Comparison between efficient coding and IB. Mutual information, I ( R ; S ), between stimulus, S , and response, R , is decomposed into population entropy, H ( R ), and stimulus-conditioned population entropy, H ( R | S ), with strategies for optimizing each highlighted. In IB, the minimal setup is a Markov chain T − S − Y , indicating that T is conditionally independent of Y given S . A typical curve of optimal encodings is shown as a function of I ( T ; S ). The diagonal dotted line illustrates the tight upper bound on I ( T ; Y ) compared to the looser data processing inequality. The horizontal dotted line marks the maximum possible value of an optimal encoding, equal to the entropy of the relevance variable H ( Y ). Classic efficient coding maximizes mutual information between response and the stimulus generally, and does not distinguish which stimulus features are most important for driving behavior. This distinction is especially important in the retina, where slow integration time constants in rods and cones [ 8 , 17 ] have the potential to cause behaviorally detrimental visual delays. Features that allow the system to anticipate the future are therefore particularly important. This idea can be formalized using the information bottleneck (IB) framework [ 18 – 20 ] ( Fig. 1 ), where the relevance variable is taken to be the future stimulus. This choice of relevance variable naturally leads to delay compensation. Analysis of retinal ganglion cell responses using the IB framework [ 21 , 22 ] reveals that small populations of retinal ganglion cells sit near an upper bound on the maximum possible information between the future stimulus and the population response given their measured amount of information about the past. Successful applications of efficient coding and IB suggests that both may capture important aspects of retina population activity. For versions of efficient coding that do not constrain what features should be encoded, the two analyses can be complementary: efficient coding constrains how information is represented, while the IB framework determines what information is represented. In the predictive setting, the question of, “What is encoded?” depends on the choice of timescale, Δ t IB , which sets how far into the future the relevance variable extends. Previous work fixed Δ t IB , but leaving Δ t IB to be inferred from the population response across different stimulus conditions establishes a novel way to examine how the retinal encoding changes with changing stimulus statistics. This idea of merging two important ways for a system to perform well is examined in the Deterministic Information Bottleneck [ 23 ]. In this framework, there are two axes of efficient coding: how much information there is between the relevance variable and the response (classic IB) and how much the entropy in the encoding distribution contributes to information between the input and response (classic efficient coding). These correspond to optimality in channel coding and source coding, respectively. As an example of why both ideas are important in sensory encoding consider the following example: two neural populations contain the same information between their response and past and future stimuli, but one achieves this with sparse spiking while the other uses very high firing rates. From an IB perspective these encodings are equivalent, yet efficient coding theory, and the Deterministic IB, would favor the sparser solution. Here, retinal population responses are evaluated along both axes: their proximity to the IB bound (for an inferred ), which captures what aspects of the stimulus are encoded, and their efficiency in representing the stimulus. This analysis is done across a range of stimulus statistics to examine if the that best describes the retina population response adapts to changing stimulus statistics. For retinal populations, the optimal increases as a function of the stimulus time constant; across all stimulus conditions the population decoding performance is well described by IB theory. RESULTS What is the appropriate relevance variable? Teasing apart how stimulus features affect predictive encoding requires a rich, yet tractable input motion model. A moving bar stimulus whose dynamics are generated by a stochastically driven damped harmonic oscillator ( Fig. 2a ), is a physically motivated choice of dynamical system that has exactly solvable correlation functions [ 24 ] while still displaying rich behaviors useful for probing the retina, like motion reversals ( Fig. 2a ). In addition to the correlation functions being exactly solvable, the IB problem for future information can also been solved exactly [ 21 ] across all choices of stimulus parameters [ 25 ]. Download figure Open in new tab FIG. 2. Connection between motion anticipation and the predictive IB. A ) A stochastically driven critically damped harmonic oscillator controls the position of a white bar on a black screen. The dynamics include both a predictable component (oscillator) and an unpredictable component (white noise drive). Example position and velocity traces are shown along with the corresponding phase space. Concentric ellipses in phase space show level sets for 1, 1.5, 2 and 2.5 σ . This stimulus design provides useful features for probing retinal responses to events such as motion reversals. B ) Schematic illustrating the effect of different measurements of position and velocity information. One-standard-deviation ellipses represent prior, present measurement, and future uncertainty. In the top row, a more predictive measurement is taken, so the future uncertainty ellipse relaxes more slowly to the prior than in the bottom row. C ) Time of maximum position r 2 for different IB optimal encodings. Increasing θ IB (equivalently, Δ t IB ) shifts the time of peak correlation with stimulus position forward, corresponding to greater amounts of motion anticipation. Total motion anticipation is defined as the time of maximal position r 2 . D ) Example trajectory showing how the IB encoding anticipates stimulus position. The stimulus position (solid line) is preceded by the IB compression (dashed line). Applying IB to the stimulus prediction problem requires choosing how far out in the future to predict. Sachdeva et al. [ 25 ] demonstrated that when the relevance variable is the future stimulus state (position and velocity) at time t + Δ t IB , the optimal encoding smoothly transitions from encoding position information (small Δ t IB ) to incorporating increasing amounts of velocity information (large Δ t IB ). This creates a family of optimal encodings as Δ t IB varies from near zero to infinity. In the high-compression regime, the two-dimensional state space is projected down to one dimension, and the optimal encoding is a linear combination of position and velocity with some Gaussian noise injected ξ . IB optimal encodings span a substantial fraction of all possible linear combinations of position and velocity ( Fig. 2b ). The space of potentially optimal encodings grows even larger when alternative relevance variables are considered, such as future velocity alone rather than the full future state. In the extreme, one could even construct an artificial relevance variable that makes any given retinal encoding match the IB-optimal one by choosing the relevance variable to match exactly what the retina encodes. This emphasizes how the IB optimum is often matchable to neural data for a careful choice of relevance variable, revealing what computations the brain may be optimized for. To address the large number of potentially optimal encoding maps, analysis of the retina population response begins by constructing the entire set of predictive IB encodings, which vary as a function of Δ t IB . Then, depending on which Δ t IB best describes the population response, that becomes the relevance variable. This extends IB from something that is about optimal encoding of information, to a tool that can be used to learn something about the population response. The that best describes the population response suggests that particular value is what the retina may be optimized to predict. It would be reasonable to conclude that two different retina populations that were well described by two different values of may have different output functions. The feature that the retina is encoding best in response to a stimulus may be considered an answer to the question, “What is the retina computing?” Motion anticipation and IB IB is an information theoretic tool, however for the choice of stimulus it has a straight-forward interpretation in terms of motion anticipation. Motion anticipation in the retina has previously been probed with deterministic moving bar stimuli, traveling at constant velocity [ 26 – 28 ] and constant speed with motion reversals [ 29 , 30 ]. The effect can be extended to a more ethologically relevant, stochastic moving bar stimulus [ 21 ]. For the deterministic moving bar stimulus, motion anticipation is defined as the location in space where the retina response is peaked relative to the leading edge of the bar stimulus, itself. For a stochastic system, a probabilistic analogue of motion anticipation is needed. Probabilistic motion anticipation is defined as the time of peak correlation between the compressed stimulus and the stimulus position. When analyzing response data from a population of retinal ganglion cells, motion anticipation is defined as the time of peak decoding performance for the population response at time t and the stimulus position at time t + Δ t . Positive Δ t is an anticipatory response. The solution of the IB problem for the stimulus shows that as the relevance variable Δ t IB moves further into the future, the optimal encoding shifts the peak correlation with stimulus position to progressively earlier times relative to the stimulus, corresponding to the response leading the stimulus by a greater amount of time. The moving bar evolves in a two-dimensional statespace, each point in time is associated with a position and a velocity. For the heavy amount of compression in the retina, the IB encoding is a noisy linear map from the two-dimensional state-space down to one-dimension. The one-dimensional retinal encodings are mapped onto unit vectors in the position, velocity plane and are parameterized by θ IB . θ IB = 0 rad corresponds to a pure encoding of position, θ IB = π/ 2 rad corresponds to a pure encoding of velocity ( Fig. 2b ). Taking τ = 1 and for unit position variance, there are the following correlation functions The correlation functions for the compression have the following correlation functions which can be used to calculate fraction of position and velocity variance captured by the IB compression and are used instead of correlation ( r x , r v ) with position and velocity to maintain consistency with a population decoding analysis. Differentiating with respect to t , a critical point is found that corresponds to a maximum at When θ IB is restricted to be between 0 (position-only encoding) and π/ 4 (position-and-velocity-equal encoding), the point in time that the compression’s correlation with position attains a maximum value that increases with θ corresponding to increasing Δ t IB in an IB problem ( Fig. 2b,c ). This gives a very clear picture of the effect Δ t IB has in the IB optimal solution. As Δ t IB is increased, the encoding displays larger and larger amounts of motion anticipation. This can be seen straightforwardly in ( Fig. 2b,c ), where the plotted encoding appears as a leading estimate of the stimulus position ( Fig. 2c ). Motion anticipation [ 26 ], a classic effect observed in the retina, appears as the IB optimal encoding of a stochastic stimulus. With this particular choice of stimulus, it is possible to evaluate the cost of motion anticipation in terms of degraded performance in stimulus position estimation. Plugging in the time of maximum position decoding performance , demonstrates that more motion anticipation corresponds to less possible position decoding ( Fig. 2c ). This occurs because any prediction of the future stimulus will be less accurate than a measurement of the present stimulus. However, motion anticipation also degrades decoding performance at the time of the measurement. This occurs because motion anticipation comes from trading position information for velocity information. For fixed amounts of channel capacity, increases in velocity information must come at the expense of decreases in position information. This can be seen in ( Fig. 2c ), where the noiseless optimal encoder at time 0 decreases, as θ IB goes from 0 to π/ 4. Evaluating the optimality of retinal response to stimuli with different time constants With this framework in hand, a population of 53 axolotl retinal ganglion cells is analyzed, as it responds to a stochastically driven critically damped harmonic oscillator with 5 different choices of time constant τ (25 ms, 50 ms, 100 ms, 200 ms, 400 ms) ( Fig. 3a,b ). Response are assessed for optimality under both efficient coding and IB. Download figure Open in new tab FIG. 3. Relative priority of position and velocity information changes with stimulus time constant A) Example bar position traces for different time constants. Noise is adjusted to keep bar position variance constant. B) Corresponding bar velocity traces. With position variance fixed, velocity variance decreases as the time constant increases. C-E) Mutual information between one bin (16.667 ms) of neural activity and ( C ) the full stimulus, ( D ) position, and ( E ) velocity. Average response delay remains consistent for total information ( C ). As stimulus time constant increases, the time of maximum position information shifts forward ( D ), reflecting increased velocity information ( E ). Red dotted line: time of peak mutual information between response and stimulus averaged across conditions. Black dotted line marks time 0. F) Two measures of coding efficiency. (blue) quantifies noise in the population response. (green) measures stimulus decorrelation, defined as the ratio of measured population entropy to that of a rate-matched independent population. G) Time to peak position information as a function of stimulus time constant. Because responses carry velocity information in all conditions, peaks are consistently later than total information peaks. The peak shifts further forward with increasing time constant, extending into the future for the τ = 400 ms condition. H) Decomposition of total mutual information into position and velocity contributions as a function of time constant. Increasing time constant emphasizes velocity information, enabling greater motion anticipation. The mutual information is measured between the population spiking activity at a give time, t , and the position and velocity of the stimulus across a range of relative times in the past or future, I ( R t ; S t ±Δ t ). All stimulus conditions show a consistent response delay, measured as the time of peak mutual information between the stimulus and response ( Fig. 3c ). While mutual information between the population response and the full stimulus (position and velocity) has a consistent time of peak information ( Fig. 3c ), there is not a consistent peak in position-only information ( Fig. 3d ). As the time constant of the stimulus grows larger, the time of peak position information moves farther and farther forward in time ( Fig. 3d ). For the slowest time constant stimulus ( τ = 400 ms), the population response becomes anticipatory ( Fig. 3g ). That is, the time at which the population best decodes the bar position occurs in the future. This happens because the neural population response carries more and more information about the velocity of the stimulus, allowing for extrapolation ( Fig. 3e ). This increase in the time of peak position decoding performance as the time constant of the moving bar increases is consistent with increasing in the predictive IB problem. As the time constant of the stimulus grows, more information is carried about velocity, and less about position ( Fig. 3h ). The application of IB makes clear how the population response of the retina changes as a function of stimulus parameters. A different measure of optimality needs to be considered to evaluate the population response in terms of informational efficiency. First, the amount of noise in the system is quantified by . This measure will be 1 for a system with no noise, and 0 for a system with only noisy responses. It corresponds to maximizing information between the stimulus and response by minimizing the conditional entropy, H ( R | S ). In the retinal data, the population response has a maximum in efficiency for τ = 50 ms ( Fig. 3f ). Second, efficient coding in terms of stimulus decorrelation is measured by the ratio of actual system entropy H ( R ) to the entropy of a rate-matched independent population H ( R ind ). This ratio, will be 1 for a fully independent population, and decay to 0 as correlations between neurons grow stronger. This measure is appropriate for comparison across stimulus conditions because stimuli were variance-matched across τ , meaning that the level of correlation between neurons should be a result of retinal response properties, rather than stimulus properties. Under this measure, a minimum in efficiency is observed for τ = 50 ms ( Fig. 3f ). Interestingly, while one measure of efficiency (noise reduction) is at a maximum, another (stimulus decorrelation) is at a minimum in these data. It is important to consider the relative magnitude of these competing effects. For the stimulus decorrelation measure , in order of increasing τ , a reduction in entropy of .58 ± .06 bits, .72 ± .03 bits, .72 ± .03 bits, .63 ± .05 bits, and .48 ± .06 bits is observed. Measuring the information lost to noise, H ( R | S ), we observe 3.8 ± .26 bits, 4.36 ± .12 bits, 4.82 ± .18 bits, 4.81 ± .44 bits, and 4.66 ± .26 bits (stimulus conditions in same order). While decorrelation of the stimulus has the potential to improve information between stimulus and response, it appears that for the level of noise in the retina, much larger gains are possible with noise suppression. The τ = 50 ms stimulus condition achieves maximal mutual information between stimulus and response ( Fig. 3c ), while still having the second lowest entropy of all stimulus conditions, and the least-independent population response. The superior performance over other stimulus conditions is a result of less noise in the population during this input drive. This suggests an interesting response regime for the retina in which correlations between retinal ganglion cells are (at the very least) not harmful, if not beneficial, for stimulus encoding. While this analysis is carried out for single time points in the stimulus and response, retinal ganglion cells populations carry information about the stimulus across time, which can have important implications for retinal coding [ 31 ]. The same analysis was performed for response vectors in time, up to 500 ms in duration and similar effects are observed (data not shown). By analyzing decoding performance across different relative time lags it is possible to assess agreement with an IB-optimal decoder. There are two free parameters in this analysis: β parametrizes the amount of information between the past and the response, and Δ t IB sets the target prediction window, the relevance variable in the IB problem. As the ratio and total amount of position and velocity mutual information changes across stimulus statistics ( Fig. 3C-E ) the IB problem that best describes the population response will also be different. Allowing the parameters to change across stimulus conditions, the IB-optimal position decoder closely matches the performance of a decoder trained on retinal population responses across relative lags for all stimulus conditions. ( Fig. 4 ). When an extra degree of freedom is allowed for how far out into the future the population of retinal ganglion cells is predicting, it is possible to observe near optimal IB encoding across a range of stimulus conditions. A fixed choice of IB relevance variable would not have described the retina response to all stimulus conditions well. This suggests a novel approach to studying adaptation effects on encoding at a high level. Download figure Open in new tab FIG. 4. IB encodings accurately describe population responses. IB encodings accurately describe population responses. Position r 2 from population activity is compared to theoretical upper bound on performance predicted by the past-future IB problem. The only free parameters are the system noise level and the predictive time offset in the relevance variable. Across all stimulus conditions, agreement between theory and data is strong. Colored vertical bars show the time of maximal correlation with bar position for all stimulus conditions. Implications of efficient coding with surprise One prediction made by efficient coding theory, particularly predictive coding [ 32 – 34 ], is that neurons encode the novelty of their inputs. The dynamic stimulus analyzed here requires a careful treatment to quantify surprise in a probabilistic sense. While many notions of statistical surprise exist [ 35 ], a common and well-motivated choice is the negative log-probability of the input - essentially how ‘rare’ an input is. This conception of stimulus novelty was found to describe the firing patterns of individual retinal ganglion cells in response to a full field flash stimulus designed to evoke the so-called Omitted Stimulus Response [ 36 , 37 ]. Estimates of the population response log-probability are used to extend this quantitative analysis explored with single cells as in [ 37 ] to the entire population. The negative log-probability of the retina population response is computed and compared to the negative log-probability of the stimulus, putting both input and output ‘surprise’ in direct comparison ( Fig. 5a ). Given that feature selectivity is shaped by the relevant time scale of prediction (Δ t IB in an IB problem), the log-probability of the stimulus is computed for the value of ( Fig. 2b ) that best describes the decoding performance ( Fig. 4 ). The stimulus-response offset chosen to compare surprise is the same as that used in the IB analysis. This leaves no free parameters, and a head-to-head comparison of the stimulus and response surprise can ben assessed ( Fig. 5a ). Download figure Open in new tab FIG. 5. Surprise encoding and the retina reversal response. A) Schematic of the procedure for comparing stimulus surprise with response surprise. Stimulus compressions retain a degree of freedom, represented as a linear combination of position and velocity parameterized by θ IB . B) Relationship between stimulus surprise and response surprise for a single trial across stimulus conditions. Trial r 2 values: 0.32 for τ = 25 ms, 0.40 for τ = 50 ms, 0.39 for τ = 100 ms, 0.34 for τ = 200 ms, and 0.29 for τ = 400 ms. C) Probability of reversal as a function of position and velocity for τ = 400 ms and Δ t rev = 800 ms. D) Probability of reversal when projected onto the stimulus compression for IB optimal (blue) and non-anticipatory θ (orange. E) Compression surprise as a function of compression value, showing the same general form as in D for the IB Optimal value of , but not the non-anticipatory θ . Reversals correspond to surprising events for an IB optimal compression. F) Cartoon illustrating this similarity. To estimate the log-probability of the population response, it is necessary to fit a model of of these collective states. A Restricted Boltzmann Machine, a flexible machine learning model that has been found to reproduce retina firing statistics [ 38 ], is an expressive model that well-captures the population response distribution. Across all stimulus conditions tested, a strong agreement between stimulus and response log-probabilities is observed, indicating that a surprising input leads to an especially rare and surprising response that could potentially be used to signal surprise to downstream circuits. Traces shown in ( Fig. 5b ) are for the IB analysis offset, and the IB-optimal , used to define the stimulus compression . This ability to decode stimulus surprise directly from population surprise demonstrates that the efficient coding theory prediction that rare responses should occur with rare stimuli is borne out across stimuli conditions. This observation suggests a tight a connection with the retina reversal response [ 29 ], a phenomenon observed in retinal ganglion cells where the sudden reversal of a moving bar evokes a large and synchronous volley of spikes. Since the statistics of the stimulus are known, it is possible to compute the stimulus reversal probability as a function of position and velocity ( Fig. 5c ), for a particular choice of stimulus time constant, τ , and time in the future, Δ t rev . This same probability can be estimated for a compressed representation parameterized with ( Fig. 5d ) These quantities are computed for the τ = 400 ms time constant, Δ t rev = 800 ms and IB optimal . For comparison the empirical reversal probability of the compression is also computed for θ = 0, which will not show any motion anticipation. For the value of that best describes the population response, there is a strong alignment between the log-probability of the IB-compression ( Fig. 5e ), and the probability of a reversal ( Fig. 5d ). For the predictive compression , novel retina population responses are likely when the bar is about to reverse. This does not hold when θ = 0, a choice that eliminates motion anticipation ( Fig. 5d ). The effect is strong for a motion-anticipating and absent for a non-anticipating θ , showing how IB motion anticipation and the retinal reversal response are related for this particular stimulus, and possibly in general. This probabilistic interpretation of the retina reversal response depends on the choice of a stimulus with well characterized, but not overly simple, stimulus statistics. This connection between rare stimulus states and reversal probability remains with the inclusion of a monotonic nonlinearity such as the kind analyzed in [ 39 ]. DISCUSSION An information theoretic interpretation of classic retinal ganglion cell response features (motion anticipation and reversal response) using IB and efficient coding arguments shows how both of these notions of “optimal” can be simultaneously evaluated in data. IB can be used to assess what computations a system is performing, including classic notions of efficient coding. When performing this kind of analysis in the context of prediction in time, it is important to consider laws of mutual information such as the data processing inequality. A time offset can be used to account for delays in the retina response, but this offset must be chosen in such a way that respects causality. If an offset is chosen too far in the past, apparent (but, of course, spurious) violations of these laws can appear. Under choices like this, mutual information between the stimulus and response can increase from the time chosen as the offset, which would be divination. An important question is how, mechanistically, these optimal and efficient computations are carried out by populations of retinal ganglion cells. Previous work has highlighted the role of amacrine cells in motion anticipation [ 40 ], and other studies suggest that gap junctions may contribute to predictive encoding of information [ 22 ], with anticipatory effects also observed in bipolar cells, likely mediated through interactions with amacrine cells [ 22 ]. A common theme among these possible mechanisms is inhibition [ 41 ], either feedback or feedforward. Theoretical analysis has made specific predictions about speed dependence of motion anticipation in feedback and feedforward inhibition [ 41 ]. Careful measurement of motion anticipation effects across a range of stimuli, similar to what is reported here, could be useful in determining whether feedback or feedforward inhibition accounts for motion anticipation effects observed in populations of retinal ganglion cells. Another open mechanistic question is how single cell biophysics may contribute to predictive processing observed in the retina. This has been explored in phenomenological models [ 26 ], but has not been linked explicitly to single cell mechanisms. Spike frequency adaptation is a form of negative feedback observed across many different types of neurons [ 42 ], which include retinal ganglion cells [ 43 ]. Such adaptation effects have been shown to cause phase leads in single neocortical pyramidal neurons injected with sinusoidal current drive [ 44 ]. This work was not done in the context of motion anticipation, but the development of phase leads is necessary for motion anticipation as described in prospective coding [ 45 ]. For this reason, it may be possible for single retinal ganglion cells to show motion anticipation effects in isolation. While pharmacological knock-outs of circuit mechanisms have effectively removed motion anticipation from retinal ganglion cells [ 40 ], suggesting that they are necessary for motion anticipation, it does not show that they are sufficient for motion anticipation. Some amount of predictive capacity may arise from single cell properties. Such effects arising from single cell biophysics would likely play a synergistic role with other circuit based mechanisms. The IB approach outlined here may be useful in other neural systems. By constructing a family of optimal encoding distributions that come from different choices of relevance variables and finding the relevance variable that best describes the system response, one is able to discover what the system is optimized for. When analyzing a population response, if there is some parsimonious relevance variable for which the response is optimal, that variable is probably closely linked to the function of the system. While prediction of future stimulus is well-motivated as a family of relevance variables, particularly for the early sensory systems, other choices may be appropriate for different brain areas. Additionally, across species, the retina has a number of specialized functions linked to different cell types; examples include the detection of looming objects [ 46 , 47 ], and motion detection [ 48 ]. More generally, across species there is a variety of more specific, ethologically important behaviors that may require entirely different choices of relevance variables [ 49 ]. As a final point, there is exceptional value in using stochastic stimuli in sensory experiments. The stochastic, dynamic, and yet analytically-tractable stimulus used here allows for a set of information theoretic analyses that would have been ill-posed with a deterministic stimulus. In addition, this stochastic stimulus enabled connections to be drawn between the retina reversal response and motion anticipation. Stimuli with rich naturalistic spatial and temporal correlation structure not only evoke the most expressive response states in neural populations, they also allow for the discovery of what matters to the brain and what it may have evolved to encode. METHODS Retina Recording 53 retinal ganglion cells from the isolated retina of an axolotl were recorded using a multielectrode array with 30 µm spacing and 252 channels following methods described in [ 50 ]. Retina Stimulus The stimulus was a white moving bar against a black screen. The dynamics of the bar were governed by the described stochastically driven critically damped harmonic oscillator. Time scales were chosen as 25 ms, 50 ms, 100 ms, 200 ms, and 400 ms. The projection used a DMD system with a refresh rate of 120 Hz. Spikes and stimulus were both binned at 60 Hz. Stimulus conditions are interleaved and each condition has 30 different trials. Mutual information estimation We take a variational approach to the estimation of information. As we know the statistics of our stimulus are Gaussian and position and velocity are independent of each other at equal times, we can use the r 2 for position and velocity estimated from a decoder to estimate mutual information between stimulus and response. Estimates of r 2 are taken from held-out test data averaged over five-folds. Following [ 31 ] we use a linear decoder. We note that when taking this approach to measuring information between stimulus and response it is extremely important to measure mutual information for both the position and velocity. A measurement based on only position would understate the amount of information in the population; in addition, it can cause violations of the data processing inequality. For example, we observe that the peak in information between population response and the position occurs in the future for one stimulus condition, while the peak in total information remains in the past. Importantly, this approach avoids the normal issues with nonparametric estimation of entropies [ 51 ]. IB analysis In the comparison between the IB and the decoded retina response, there are two free parameters. β , corresponding to the amount of noise, and θ IB corresponding to the value of Δ t IB in the IB problem. Across all conditions we solve for these two parameters by picking two points in time and substituting these into the expression for . The number of free parameters is reduced to one by forcing agreement between the IB decoder and the neural data at time l , where l is an offset. This leaves a single parameter that determines the total amount of anticipation. This remaining parameter is fit to minimize mean squared error with the observed values, Brent’s method is used. Similar to previous work [ 22 ], we introduce an offset into the IB analysis. The offset sets the time index that we take as t when considering the decoding performance of the population at time t and position at time t + k . We chose our offset to be 16.667 ms (one time bin) from the time of maximal mutual information. The time of maximal mutual information is the minimum offset that respects the data processing inequality [ 3 ], which states that in a Markov chain X − Y − Z we have I ( X ; Z ) ≤ I ( X ; Y ). In the context of temporal processes, this reduces to the statement that it is impossible to know more about the future than is known about the past, and is necessary for respecting causality. The extra time-bin was taken as an extra precaution. This approach differs from [ 22 ], where the ofset was chosen as the time a filter from an encoding model reached a maximum value. This has no guarantee that it will respect the data processing inequality. We demonstrate the agreement between the IB optimal encoding and the neural population code using the performance of a decoder trained on spiking data. Agreement with the IB predictions is assessed across a range of relative lags Δ t between stimulus and response. This analysis emphasizes the performance across a range of values, rather than only the Δ t IB taken as the relevance variable because, while IB sets an upper bound on information available at time t + Δ t IB , it also sets expected correlations across all times. Analyzing a range of relative lags Δ t is particularly important in cases where Δ t IB becomes large. For large Δ t IB information between the compression at time t + Δ t IB and the stimulus will approach 0. As it is approaching 0 any coding approach would also give an amount of information near 0. What actually makes different choices of relevance variables distinguishable from one another often occurs well before the particular value of Δ t IB . Entropy estimation To estimate the entropy of the neural population we use the cross-entropy estimate found from inferred Restricted Boltzmann Machines. This gives an upper bound on the true entropy. These expressive models have been found to accurately describe retinal population activity [ 38 , 52 ]. The empirical cross-entropy is an unbiased estimator of the true cross-entropy and bounds the true entropy from above. The gap between the cross-entropy and the population entropy is the Kullback-Leibler divergence between the underlying distribution and the model, which is expected to be small for such an expressive model. We use five-fold cross-validation over stimulus trials and report the average entropy on held-out test data as the estimate, reported error bars are the standard deviations over folds. Folds are shuffled over trial indices instead of all data to manage contamination from temporal correlation. As Restricted Boltzmann Machines are unnormalized statistical models, we need to estimate the partition function to report the cross-entropy. We use the Good-Turing estimator to estimate the partition function [ 53 ], as discussed in [ 54 ]. For optimization we use the minimum probability flow approach [ 55 ], and use model samples with the factorization assumption to construct a connectivity matrix [ 56 ]. We used the cross-validation performance to select the appropriate number of hidden units. Model convergence was additionally assessed by comparing model covariance to data covariance, and model spike count distributions to data spike count distributions. All reported entropy measures based on these models are reported from the average of held-out test data. Error bars are the standard deviations over folds. Stimulus decoding model All stimulus decoding models are five-fold cross-validated linear regression. To manage temporal correlations, folds consist of trial shuffles, rather than shuffles over all data. All reported information measures based on these models are reported from the average of held-out test data. Error bars are the standard deviations over folds. The decoding models and Restricted Boltzmann Machines use the same folds to maintain consistency in measures involving both entropy and mutual information. Author Contributions K.B. B.L J.S O.M S.E.P. designed research; K.B. B.L J.S O.M. and S.E.P. performed research; K.B. analyzed data; and K.B. and S.E.P. wrote the paper. Competing Interest Statement The authors declare that they have no conflicts of interest. Acknowledgments KB would like to thank Tobias Kühn for useful discussions, and Sophie Colt for helpful comments on the manuscript. 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