Efficient Neural Codes that Minimize L p Reconstruction Error

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1 1 Efficient Neural Code that Minimize L p Recontruction Error Zhuo Wang 1, Alan A. Stocker 2, 3, Daniel D. Lee3, 4, 5 1 Department of Mathematic, Univerity of Pennylvania 2 Department of Pychology, Univerity of Pennylvania 3 Department of Electrical and Sytem Engineering, Univerity of Pennylvania 4 Department of Computer and Information Science, Univerity of Pennylvania 5 Department of Bioengineering, Univerity of Pennylvania Abtract The efficient coding hypothei aume that biological enory ytem ue neural code that are optimized to bet poibly repreent the timuli that occur in their environment. Mot common model utilize information theoretic meaure, wherea alternative formulation propoe incorporating downtream decoding performance. Here we provide a ytematic evaluation of different optimality criteria uing a parametric formulation of the efficient coding problem baed upon the L p recontruction error of the maximum likelihood decoder. Thi parametric family include both the information maximization criterion a well a quared decoding error a pecial cae. We analytically derived the optimal tuning curve of a ingle neuron encoding a one-dimenional timulu with an arbitrary input ditribution. We how how the reult can be generalized to a cla of neural population by introducing the concept of a meta tuning curve. The prediction of our framework are teted againt previouly meaured characteritic of ome early viual ytem found in biology. We find olution that correpond to low value of p, uggeting that acro different animal model, neural repreentation in the early viual pathway optimize imilar criteria about natural timuli that are relatively cloe to the information maximization criterion. 1 Introduction The efficient coding hypothei tate that biological enory ytem have limited coding reource and therefore eek to employ coding trategie that are optimally

2 adapted to the tatitical tructure of their enory environment (Attneave, 1954; Barlow, 1961; Madde et al., 1985; Theunien et al., 1991; Fitzpatrick et al., 1997; Harper et al., 2004). Several tudie have experimentally demontrated that enory neural code eem to indeed follow input ditribution tatitic in order to reach higher coding efficiency (Brenner et al., 2000; Twer et al., 2001; Dean et al., 2005; Ozuyal et al., 2012). A large fraction of previou work aumed that neural repreentation are tuned to maximize the mutual information they are able to convey about the timulu value given ome overall contraint on available metabolic cot, e.g. total number of pike (Laughlin, 1981; Linker, 1989; Atick et al., 1990; van Hateren, 1993; Seung et al., 1993; Nadal et al., 1994; Brunel et al., 1998; Zhang et al., 1999; Pouget et al., 1999; Kang et al., 2004; Sharpee et al., 2006; McDonnell et al., 2008; Nikitin et al., 2009; Tkacik et al., 2010; Yarrow et al., 2012; Katner et al., 2015). Thi Infomax criterion ha been a preferred choice becaue it doe not require making any further aumption about potential downtream computation and tak the encoded timulu may be involved in. On the other hand, a few tudie have taken a downtream perpective and have argued for optimality criteria that conider how well the timulu information can actually be recontructed from the neural repreentation. They often ue a metric criterion in term of the mean quared recontruction error (Bethge et al., 2002, 2003; Beren et al., 2009; Yaeli et al., 2010; Doi et al., 2011). Thi recontruction metric ha been hown to optimize performance in perceptual etimation and claification tak (Salina, 2006). Recently there have been increaing interet in comparing the information with the metric approach (Ganguli et al., 2010; Gjorgjieva et al., 2014; Grabka-Barwinka et al., 2014). However, a unified comparion and evaluation of thee different approache i currently lacking. Here, we provide a unified framework to compare thee optimal criteria. We introduce a parametric formulation of the efficient coding problem in term of minimizing the overall recontruction error according to the L p norm, a a function of the norm parameter p. We aume recontruction from a maximum likelihood etimate (MLE) decoder in the aymptotic time limit. More pecifically, we conider a one-dimenional timulu with ditribution f() that i encoded with tuning curve() h() for m neuron(). While the mapping h() i determinitic, we aume the neural repone r to follow a ditribution P (r h()) according to neural noie. For both Poion and Gauian noie, we analytically derive the optimal tuning curve h to achieve minimal L p mean recontruction error for arbitrary timulu ditribution. Thi framework include both the Infomax a well a mean-quared error optimal olution in the limit of p 0 and p = 2 repectively. We firt focu on olution for the optimal tuning curve h() of a ingle (igmoidal) neuron encoding the timulu. We then how how the ingle neuron tuning curve olution can be naturally extended to population of neuron. Under certain aumption, the optimal ingle neuron tuning curve h() can be related to an optimal meta-tuning curve of the neural population, from which the individual tuning characteritic of the population of neuron can be determined. In the context of thi theoretical framework, we invetigate how known tuning characteritic of biological enory ytem can be explained. We compare the meaured tuning characteritic of early enory repreentation in the fly, the cat, and the monkey for known timulu tatitic with prediction from our framework. For the example we teted, the biological tuning characteritic are quite well predicted by our frame- 2

3 P (r h()) encoding decoding L p lo ŝ(r) p r Figure 1: Efficient coding problem in term of recontruction error. A one-dimenional timulu i encoded in a neural repone pattern r. We define the optimal tuning curve h() a the one that minimize the overall L p recontruction error according to an MLE decoder. We tudy how the optimal coding trategy i dependent on the norm parameter p. The Infomax olution i equivalent to the optimal encoder for p 0. work, and are bet matched for mall value of the norm parameter p. We conclude that early enory repreentation in biological ytem may be optimized to convey maximal information. 2 Optimal Neural Coding for a Single Neuron We tart with the cae where a ingle neuron i encoding a one-dimenional timulu variable. We aume that follow a ditribution denity f(). We alo aume that the neuron average firing rate i determined by a igmoidal function h(). The actual oberved firing rate r i ubject to neural noie, whoe variability i decribed by a tochatic model P (r h()). We do not limit the noie to be defined by canonical Poion piking model. Rather, we only aume that (a) the mean firing rate i equal to the output of the tuning curve r = h() and (b) the pike generating proce i independent from the neuron piking hitory. With ufficient encoding time or with independent obervation of identical neuron, the accumulated noie i aymptotically normal with zero mean and fixed variance according to the Central Limit Theorem. In order to decode the input timulu, we take the maximum likelihood etimator (MLE) ŝ(r), which i aymptotically unbiaed and efficient (Cover et al., 1991). In order to find the L p optimal tuning curve for a one dimenional timulu, we need to minimize the mean L p lo of the maximum likelihood etimator. The only contraint for the igmoidal tuning curve i the aturation limit of the firing rate. Within the regime of low noie limit, the maximum firing rate doe not affect the optimality. Therefore we aume 0 h() 1 without lo of generality, which lead to the optimization problem minimize ŝ(r) p,r (1) ubject to 0 h() 1. (2) 3

4 2.1 Objective Function in term of Fiher Information To get inight into the optimization problem, we analyze the Fiher Information. The Fiher information I() decribe the preciion of the bet poible etimator for each pecific individual timulu. For any, I() can be calculated according to it definition ( ) 2 I() = log p(r ) (3) where the conditional ditribution p(r ) decribe the tochatic neural repone for a given timulu and the average i taken over r but not. It ha been hown that in the aymptotic limit of long encoding time, the total Fiher information characterize the preciion of the etimator ŝ in recontructing the timulu (ee Appendix A.1) (ŝ(r) ) Normal(0, I() 1 ) (4) ŝ(r) p r = cont(p) I() p/2 (5) It i clear from Equation 5 that larger Fiher information lead to maller L p error. One example i the Cramer-Rao lower bound when p = 2. The more general Eq.(5) etablihe the connection between L p lo in Eq.(1) and the Fiher information. Thi reult in an equivalent optimization in term of Fiher information: r minimize I() p/2 (6) In addition to the L p -error minimization problem, we alo conider the well-known Infomax optimization which maximize the mutual information between the repone and the timulu. It ha previouly been hown that Fiher information can be related to mutual information Brunel et al. (1998). In our framework, Infomax i equivalent to optimizing the logarithm of Fiher information: I(r, ) = 1 2 log I() + cont (7) minimize log I() (8) 2.2 Contraint in term of Fiher Information Next we how how to incorporate contraint in Eq.(2) into the ame framework. For a one dimenional timulu variable, the Fiher information of a neuron i fully determined by the nonlinear tuning curve h() and the noie model. Here we how the reult for both Poion noie (P) and contant Gauian noie (cg), with detail provided in the Appendix A.2. P: I() h () 2 (9) h() cg: I() h () 2 (10) 4

5 Thee formulation can eaily be inverted for any given Fiher information allocation I(), the correponding nonlinear tuning curve i ( P: h() cg: h() Given bound contraint on the tuning curve in Eq.(2), we have P: cg: I() d I(ξ) dξ ) 2 (11) I(ξ) dξ (12) h () d = 2 h() cont (13) h() I() d h () d = h() cont (14) Ignoring irrelevant contant calar term which do not affect the optimal form, thee contraint can be unified: P or cg: ubject to I() d cont (15) Since it i alway better to have more Fiher information, equality in Eq.(15) mut hold for optimality. To ummarize, the objective function in Eq.(6) attempt to optimally allocate the Fiher information I() acro the pace of the timulu variable with ditribution f() under the integral contraint in Eq.(15). After determining the optimal allocation I (), the optimal nonlinearity h () can then be determined uing Eq.(11) or Eq.(12), depending upon the neural noie model. 2.3 Single Neuron Reult According to the above analyi, olving the L p recontruction error minimization problem i equivalent to olving the Fiher information allocation problem. For each p value in the L p -minimum decoding lo criterion, the optimization problem i minimize (I()) p/2 = f() (I()) p/2 d (16) ubject to I() d cont (17) Thi variational problem can eaily be olved and the optimal olution i ) 2 I () f() 2/(1+p) (18) P: ( h () = dξ f(ξ)1/(1+p) dξ (19) cg: h () = f(ξ)1/(1+p) dξ f(ξ)1/(1+p) dξ (20) 5

6 A imple comparion between the two noie model reveal that the optimal tuning curve for a neuron with Poion noie i exactly the quare of the optimal tuning curve for a neuron with contant Gauian noie. Thi relationhip wa firt reported by Bethge et al. (2002) and Johnon et al. (2004). The quaring tranformation how that the optimal coding under Poion noie tend to utilize more reliable low firing rate rather than more unreliable higher rate. Below we focu on the contant Gauian noie olution and dicu the link between our general formula and everal reult that have been previouly reported in the literature: When p = 0, the L 0 -minimum olution i given by the cumulative function of the input ditribution, h () f(ξ) dξ. (21) When p = 2, the L 2 -minimum olution i given by the cumulative function of the cube root of the input ditribution, h () f(ξ) 1/3 dξ (22) When p, the optimal tuning curve h () converge to a linear function becaue it derivative approache a contant function of and the prior p() i no longer relevant. However thi uually require the timulu to be bounded [ min, max ] otherwie the integral of f() 1/(1+p) will diverge for ufficiently large p. Note that optimizing the L p -min problem Eq.(16) when p 0 lead to the ame optimal olution a the Infomax problem in Eq.(8). Thi olution, firt propoed in (Laughlin, 1981; Nadal et al., 1994), i known a the output equalization rule becaue the output h () i uniformly ditributed within it range limit. We will informally refer to both L 0 -min and the Infomax olution in the remainder of thi paper. When p = 2, the optimal olution in Eq.(22) minimize the mean quare error of the recontructed timulu. Thi olution wa firt propoed for optimal RGB color perception (Twer et al., 2001) and dicued in (Wang et al., 2012). To ummarize, the olution in Eq.(20) provide a ytematic undertanding of the optimal nonlinearitie for the variou criteria a a function of by p. In Figure 2 we illutrate different L p optimal tuning curve for a tandard Gauian timulu prior. Intuitively, the efficient coding problem can be undertood a optimizing the allocation of neural decriptive power acro an inhomogeneou timulu ditribution. Depending upon the value of p, the optimal allocation trategy balance between more frequently appearing timuli with le frequent one. Strategie correponding more with Infomax (p near 0) emphaize timuli with higher likelihood of appearing. On the other hand, L p - optimal trategie with large p are more conervative and need to pend more reource to encode more urpriing timuli ince the error penalty i larger. 6

7 a denity f() timulu prior timulu b optimal Fiher Information c p = 0 I () f() 2/(p+1) p = 2 I () f() 2/(p+1) p = 8 I () f() 2/(p+1) Poion Gauian optimal igmoidal tuning curve h() h() h() Figure 2: The L p optimal igmoidal tuning curve for for p = 0, 2, 8 for both Poion or contant Gauian noie model. (a) the Gauian timulu ditribution (prior). (b) for each p, the optimal Fiher Information I () i derived baed on the prior ditribution (c) The optimal tuning curve for Poion noie (blue line) or contant Gauian noie (red line). 2.4 Example of Variou Stimulu Prior Ditribution We derived optimal tuning curve for a few example timulu ditribution (prior). In particular, we conidered prior ditribution that can be expreed a generalized Gauian ditribution with cale parameter c and hape parameter β. For implicity we only how olution for the contant Gauian noie aumption. From Eq.(20), the L p -optimal tuning curve i related to the input timulu ditribution: f() exp ( c β) (23) ( ( ) ) β h () f() 1/(1+p) exp c. (24) (1 + p) 1/β Therefore for a certain value of p, the nonlinearity i imply a recaled verion of the cumulative function of f(). The calar (1 + p) 1/β i a decreaing function of β. In Figure 3 we illutrate three different cae: in the extreme of uniform ditribution cae where β =, the calar remain a contant and there i no difference acro all the L p -optimal tuning curve; for the Gauian ditribution cae where β = 2, the calar grow ub-linearly a (1 + p) 1/2 ; for the Laplacian ditribution cae where β = 1, the calar grow linearly a (1 + p). Another important concluion we would like to highlight i that all the L p -optimal olution except L 0 are not invariant under nonlinear timulu tranformation. For example, the L 2 -optimal olution for a poitive valued timulu i not identical to the L 2 -optimal olution for the ame timulu tranformed to a logarithmic cale. The L 0 - optimal olution i the only olution that i invariant under any one-to-one timulu tranformation (Cover et al., 1991). Thi fact again demontrate the intuition that L p -min trategie are highly tak-driven the olution change if the timulu variable undergoe ome nonlinear tranformation before being proceed. 7

8 a denity f() Uniform ditribution timulu b optimal igmoidal tuning curve p = 0 h() p = 2 h() p = 8 h() c denity f() Gauian ditribution timulu d optimal igmoidal tuning curve h() h() h() e denity f() Laplacian ditribution timulu f optimal igmoidal tuning curve h() h() h() Figure 3: The L p optimal igmoidal tuning curve of a ingle neuron with contant Gauian noie model. Here we compare the reult for variou form of prior ditribution: uniform ditribution (a)-(b), Gauian ditribution (c)-(d) and Laplacian (or double exponential) ditribution (e)-(f). 3 Generalization to Neural Population The tuning characteritic of optimal neural population code have been tudied (Zhang et al., 1999; Pouget et al., 1999; Kang et al., 2004; McDonnell et al., 2008; Nikitin et al., 2009; Ganguli et al., 2010; Yaeli et al., 2010). The concluion from thee tudie i that the olution largely depend on the individual aumption made in the correponding derivation. 3.1 Neural Population Aumption Certain aumption are neceary to derive a well-contrained optimization problem. Rather than allowing all neuron in the population to independently exhibit arbitrary nonlinear tuning curve, we aumed the tuning curve of the k-th neuron to have the following form h k () = h 0 (ψ() ψ( k )) (25) We refer to ψ() a the meta-tuning curve that tranform the timulu to neural pace. For each neuron, k i the characteritic timulu aociated with that neuron. For example, k can be the preferred timulu (at which the neuron elicit maximum neural 8

9 repone) for neuron with unimodal tuning curve or the emi-aturation timulu (at which the neuron elicit half of maximum neural repone) for neuron with igmoidal tuning curve. Below we denote = ψ() and k = ψ( k ) reulting from the output of the metatuning curve. Further aumption are: (a) All neuron in the population hare the ame given nonlinearity h 0 ( k ). (b) The characteritic timuli k are uniformly ditributed, in other word the pacing = k k 1 between adjacent neuron i a contant. (c) h 0 and h 0 are lowly varying when meaured at the cale of, i.e. h 0 ( k ) h 0 ( k + ) and h 0( k ) h 0( k + ). When i mall, thi contraint i equivalent to h 0 and h 0 being continuou. (d) The neuron have independent output noie o the total Fiher information of the population i the linear um of each individual one I total () = k I k() (ee Appendix A.3 for proof). Thee aumption are ometime referred to a the uniform tiling propertie of a neural population (Ganguli et al., 2010; Grabka-Barwinka et al., 2014). It i important to note that the aumption (a) and (b) limit the olution to a ub-pace of all poible population code for which the mapped timulu i encoded by a homogeneou population (ee Figure 4). In our model, the total Fiher information of the population with either the Poion noie or contant Gauian noie (ee Eq.(9) or Eq.(10)) become: I 0 I total ( ) = k I k ( ) = k h 0( k ) 2 h 0 ( k ) or h 0( k ) 2 (26) The form of h 0 ( ) i fixed and often aumed but not limited to be either unimodal or igmoidal. In Figure 4 we illutrate how to determine the individual tuning curve of the inhomogeneou neural population. 3.2 Optimal Meta-tuning Curve For any meta-tuning curve = ψ(), we can calculate the Fiher Information of the k-th neuron and the total Fiher information for the population, with repect to the original timulu a P or cg: P: I k () h 0(ψ() k ) 2 h 0 (ψ() k ) ψ () 2 (27) cg: I k () h 0(ψ() k ) 2 ψ () 2 (28) I total () = k k I k () I 0 ψ () 2 (29) In the population coding cae, the mean L p recontruction error of i related to the total Fiher information and we need to minimize the following term (I total ()) p/2 = f() (I total ()) p/2 d (30) 9

10 a warped timulu homogeneou population firing rate meta-tuning curve ψ() b warped timulu homogeneou population firing rate meta-tuning curve ψ() inhomogeneou population inhomogeneou population Figure 4: Under our aumption, the inhomogeneou neural population tuning i derived by warping a homogenou tuning decription through the meta-tuning curve, i.e. the timulu pace i nonlinearly tranformed according to the meta-tuning curve. via the igmoidal meta-tuning curve ψ(). Two repreentative choice of h 0 are (a) unimodal and (b) igmoidal. where f() i the prior ditribution of the timulu. We can limit the output of a non-decreaing meta-tuning curve to the range 0 ψ() cont. Then minimizing the L p recontruction error i equivalent to the following optimization in term of the meta-tuning curve ψ(): minimize (I total ()) p/2 I p/2 0 f()ψ () p d (31) ubject to ψ () d cont. (32) Thi optimization problem i the ame a the contant Gauian noie cae we previouly dicued in Section 2.3. Thi lead to a olution for the optimal meta-tuning curve ψ () with correponding total Fiher information: ψ () f() 1/(1+p), Itotal() f() 2/(1+p) (33) ψ () = f(ξ)1/(1+p) dξ (34) f(ξ)1/(1+p) dξ Thi reult illutrate that under our model, the Fiher Information allocation for the population i entirely determined by the meta-tuning curve ψ(), in the ame way a the Fiher information allocation i determined by the igmoidal tuning curve h() of a ingle neuron with contant Gauian noie. In Figure 5 we how the L 0, L 2 and L 8 optimal neural population for encoding a timulu variable with a Gauian ditribution. Compared to previou work by Ganguli et al. (2010), our framework conider a more contrained cla of neural population becaue it aume a fixed gain acro neuron. Our formulation, however, allow u to pecify an entire family of L p -optimal olution that moothly incorporate the pecial cae of the Infomax and the MSE olution. 10

11 a denity f() timulu prior timulu b optimal metatuning curve p = 0 = ψ () p = 2 = ψ () p = 8 = ψ () c h k () h k () h k () unimodal population d h k () h k () h k () igmoidal population Figure 5: The L p optimal neural population for p = 0, 2, 8 and a Gauian timulu ditribution. Panel (a), (b) are replicated from Figure 2 and the optimal meta-tuning curve for the population i identical to the optimal tuning curve of a ingle neuron with contant Gauian noie. Here we how two different kind of optimal neural population, where each neuron ha (c) unimodal tuning curve or (d) igmoidal tuning curve. 4 Relaxing the Aymptotic Aumption For both the ingle neuron cae and the neural population cae, our reult o far have relied on everal key aumption. The mot retrictive one i the aumption that neuron are operating in the aymptotic long time limit. In thi limit, the optimal decoder naturally converge to the maximum likelihood etimator. In contrat, in a more realitic cenario where encoding time i hort, it i generally the cae that a Bayeian (and uually biaed) decoder will perform better. Unfortunately it i difficult to derive analytic olution in thi cae yet numerical effort have been made (Bethge et al., 2003; Nikitin et al., 2009). Furthermore, the derivation of the optimal Bayeian decoder can be intractable for arbitrary prior ditribution. In order to provide a ene of how well our derived analytic olution hold for horter encoding time, we compared their predicted performance to the actual meaured performance obtained by numerical imulation. The decoding performance of our L p optimized coding olution can be eaily imulated for arbitrary encoding time. For reaon of implicity, we conidered a tandard Gauian timulu ditribution p() in our imulation. The encoding proce i traightforward: timuli are ampled and encoded by the L p optimal code with additional Poion piking noie. For the decoding proce, we examined both the aumed unbiaed, maximum-likelihood etimator 11

12 (MLE) and the maximum a poteriori etimator (MAPE). In both cae, iterative gradient decent method (Newton method) wa ued to find the timulu with maximal likelihood (for MLE) or maximal poterior likelihood (for MAPE). The mean L p decoding error wa then calculated over a large et of generated timuli and compared to the theoretical prediction. For a neuron with maximum firing rate r max and a fixed length of the time window T, the key variable i the maximum allowed pike-count N max = r max T. For each value of N max we run a total of 100 independent trial and in each trial, 100,000 timuli were randomly generated. Thi experiment wa done for both a ingle neuron with igmoidal tuning curve and for a population of neuron with unimodal tuning curve. Reult are hown in Figure 6. A expected, the theoretical prediction were more accurate when N max wa large, with the critical value for N max increaing a a function of p. For horter encoding time, our reult how that the MAPE i a better etimator depite the imilar performance for larger N max. The performance of the MLE eem to be lower bounded by our theoretical prediction (ee the olid line) but the MAPE benefit from the prior information and i upper bounded by a contant related to that prior. In the ingle neuron cae, the critical pike-count N max range from approximately 10 2 (for p = 0.01) to approximately 10 4 pike (for p = 2). For ome enory neuron, uch a the H1 neuron of a blowfly (ee Section 5.1), the maximal firing rate r max can be a high a 100Hz which mean that the critical time for the long encoding aumption to be valid i around T 1 ec (for p = 0.01) to T 100 ec (for p = 2). In the neural population cae, we run imulation with K = 11 neuron with unimodal tuning curve. A expected, the performance in term of the L p error i one order of magnitude better than for the ingle neuron cae. Correpondingly, the critical pike-count N max i much maller: from approximately (for p = 0.01) to approximately pike (for p = 2). For mall p value, the performance matche the theoretical prediction for population containing a few a 11 neuron with N max 3 pike per neuron. For larger p value uch a p = 2, thi number may increae to N max 30 pike per neuron. In um, we found that depending on the value of p the long time-limit aumption can be reaonably relaxed for hort encoding time. In particular, we find that the critical pike-count can be a low a N max = 3 30 pike per neuron which jutifie the biological relevance of our reult. Generally, the prediction of our framework are much le contrained for maller p value. We have alo found that the performance of a Bayeian decoder (the MAPE) tend to be better than the MLE decoder, which how that the optimality of our olution (MLE) trongly rely on the unbiaed aumption. Fortunately, thi limitation i ubordinated to the hort encoding time limitation. The MAPE itelf i aymptotically unbiaed and ha imilar performance a the MLE decoder once the critical N max i reached. 5 Efficient Code in Viual Perception Our theoretical analyi raie the quetion of which efficiency criterion the brain actually ue to encode information. In thi ection, we conidered everal different modalitie in early viual perception: motion encoding, orientation encoding and contrat encoding. In each cae, we attempted to etimate the prior ditribution of the input 12

13 a 1 p= p= p=2.00 log L p lo log 10 N max log 10 N max log 10 N max b 0 p= p= p=2.00 log L p lo log 10 N max log 10 N max log 10 N max Figure 6: The imulated L p encoding error (MLE: red dot, MAPE: blue cro) v. theoretical prediction auming unbiaed etimator (olid line) or uing only prior information (dahed line). The marker indicate the median over 100 trial. (a) The performance of a ingle neuron with igmoidal tuning curve (ee e.g. Figure 3d). (b) The performance of a population with K = 11 neuron with unimodal tuning curve (ee e.g. Figure 5c). The vertical axi i the mean L p lo ŝ p 1/p and the horizontal axi i N max, both in logarithm pace with bae 10. timulu and compared the tuning characteritic of the predicted efficient coding model with publihed phyiological data. 5.1 Speed Encoding by a Single Blowfly H1 Neuron We firt analyzed data from the H1 neuron of blowfly, which encode the peed of a horizontally moving bar. The analyzed dataet (van Steveninck et al., 1997) wa collected from a fly H1 neuron reponding to a tochatically generated viual motion timulu. The data wa taken for 20 minute at a ampling rate of 500Hz. We binned the neural data into 1200 bin with duration t = 1 econd and calculated the average timulu i and the number of pike N i for i = 1,..., 1200 in each bin. Thi timulurepone relation i plotted in Figure 7a. The natural peed prior for the blowfly i unknown. However, baed on the invetigation of natural movie clip, previou reearch ha propoed that the prior ditribution for viual peed hould follow a power-law function of the form f() (1 + /v 0 ) 2, where v 0 > 0 i a cale parameter (van Hateren, 1993; Dong et al., 1995). For thi particular form of the prior, the optimal L p tuning curve h p() for a neuron with Poion 13

14 noie can be analytically computed. h p () f() 1 1+p h p() ( 1 + ign() ( 1 1 (1 + /v 0 ) 1 p 1+p )) 2 (35) a firing rate (Hz) obervation data bet fit c v 0 (deg/) data likelihood optimal v0 bet fit high b prior denity timulu (deg/) extracted prior d p data likelihood bet fit likelihood low timulu (deg/) p Figure 7: (a) the timulu-repone data collected from a fly H1 neuron (van Steveninck et al., 1997) and we plot the bet tuning curve uing the parametric model in Eq.(35). (b) the predicted prior ditribution to which the fly H1 neuron i mot likely adapted. (c) the optimal parameter v 0 and p i choen to maximize the data likelihood. Dah line how the optimal parameter v 0 (p) a a function p. (d) The maximum data likelihood for each pair (p, v 0 (p)) a a function of p. It can be een that for thi parametric form of the prior ditribution, the L p optimal olution exit only for 0 p 1. In order to infer the prior ditribution and the optimal norm parameter, we optimized for the parameter v 0 and p that maximized the data likelihood. The reult i hown in Figure 7b and repreent the predicted peed prior ditribution to which the H1 neuron i optimally adapted to. In Figure 7c-d we can ee that parameter value v 0 = 21.3 deg/ec and p = 0 lead to the highet data likelihood. However other pair of (p, v 0 ) for p < 0.8 alo yield good likelihood core. 5.2 Orientation Encoding with Neural Population We alo applied our propoed framework to analyze biological neural population that encode local viual orientation. We firt etimated the prior ditribution f(θ) of local viual orientation θ from a natural image dataet (van Hateren et al., 1998) uing a filter analyi at a ingle patial cale (detailed decription in Appendix B). The reulting prior ditribution i hown in Figure 8c and i very imilar to previouly etimated 14

15 ditribution (ee e.g. Girhick et al. (2011)). Baed on the etimated prior denity, we derived the optimal meta-tuning curve ψ(θ) for variou value of the norm parameter p (ee Figure 8b). The unimodal tuning curve of the population (ee Figure 8d) were then determined a decribed in Section 3.2 auming an homogeneou population of certain tuning width w (ee Figure 8a). Below we compare prediction of the model population with meaured biophyical characteritic of orientation tuned neuron. The firt prediction i with regard to neural denity. De Valoi and colleague reported that the ratio between neuron tuned for oblique v. cardinal orientation i about 0.66 in area V1 of the macaque (De Valoi et al., 1982). In our framework the neural denity a a function of θ i directly related to the derivative of the meta-tuning curve (Figure 8f). In order to compute the ratio between the number of neuron tuned for the oblique v. the cardinal orientation, we binned the neural population into two ubpopulation hown a blue/red region in Figure 8f. The predicted ratio i a function of the norm parameter p (Figure 8e); for p 0.37 the ratio of the model population matche the ratio found for neuron in V1. We can alo predict how the tuning width depend on the preferred timulu of the neuron. Following the definition of Ringach et al. (2002), we defined the tuning width w a the length of the orientation interval over which a neuron mean repone i at leat 1/ 2 of it peak firing rate. Figure 8h how the predicted tuning width w(θ) a a function of the preferred orientation θ of a neuron in the model population. Each curve how the tuning width w(θ) for a different aumed contant tuning width w in the homogeneou population (Figure 8a). From thee continuou function we calculated the firt and third quartile w 1Q, w 3Q of the tuning width acro the inhomogeneou population. For each p value, the poible value of w 1Q ( w) and w 3Q ( w) form a curve with parameter w a hown in Figure 8g. A comparion of the quartile prediction with phyiological data from neuron in area V1 of the macaque (Ringach et al., 2002) ugget that the model bet matche the data for a norm parameter of value p = Finally, we can make prediction about tuning curve aymmetrie. Specifically, we compared the predicted aymmetry index (Henry et al., 1974) of our model population with the value found for biological neuron. Similar to the tuning width, the predicted aymmetry index i alo a function of the aumed tuning width w of the neuron in the homogeneou population (ee Figure 8j). We computed the predicted relationhip between the mean aymmetry index and the median tuning width for different p value and compared it with meaurement from imple cell in triate cortex of the cat (Henry et al., 1974). The reported median tuning width (meaured at 1/2 peak amplitude; we have rectified our prediction accordingly) of 34 and aymmetry index 1.26 matche our prediction for p 0.85 (ee Figure 8i). In ummary, we found that the meaured orientation tuning characteritic of neuron in primary viual cortex of the macaque and the cat match thoe model prediction that correpond to fairly low value of p. 15

16 a width w = 10% b p=0.0 p=0.5 p=1.0 c homogeneou pace repone d meta-tuning curve timulu (deg) timulu (deg) timulu (deg) prior ditribution image data mooth fit repone e timulu (deg) f timulu (deg) timulu (deg) timulu (deg) oblique/cardinal ratio g prediction de Valoi et al p h neural denity timulu (deg) timulu (deg) predicted denity cardinal oblique timulu (deg) 3Q tuning width (deg) i p= p=0.5 p=1.0 Ringach et al Q tuning width (deg) tuning width (deg) j timulu (deg) timulu (deg) w = 25% w = 20% w = 15% w = 10% w = 5% timulu (deg) mean aymmetry index p=0.0 p=0.5 p=1.0 Henry et al median tuning width* (deg) aymmetry index timulu (deg) timulu (deg) w = 25% w = 20% w = 15% w = 10% w = 5% timulu (deg) 16

17 Figure 8: Comparion between theoretically predicted and phyiologically meaured tuning characteritic of orientation tuned neural population. (a)-(d) cartoon example of L p -optimal neural population derived baed on a homogeneou neural population and the optimal metatuning curve, which i determined by the prior ditribution extracted from natural image. The p value are 0, 0.5 and 1. (e)-(f) the oblique veru cardinal ratio prediction i compared with previou reult (De Valoi et al., 1982) on macaque V1 foveal neuron, which ugget p (g)-(h) the 1t and 3rd quartile tuning width prediction i compared with previou reult (Ringach et al., 2002) on macaque V1, which ugget p (i)-(j) the aymmetry index and median tuning width(*) prediction i compared with previou reult (Henry et al., 1974) on cat triate cortex, which ugget p (* the tuning width here i meaured at half amplitude to be conitent with previou tudy.) 5.3 Contrat Encoding with Neural Population We alo applied our framework to make prediction for the contrat gain characteritic of neuron in early viual cortex. The contrat of natural image ha been defined in multiple way in the literature. Two tandard definition of local contrat are the root-weighted-mean-quare contrat (Najemnik et al., 2005; Mante et al., 2005) and the equivalent-michelon contrat (Brady et al., 2000; Tadmor et al., 2000; Clatworthy et al., 2003). We ue the equivalent-michelon contrat in order to match our prediction with recorded phyiological data (Clatworthy et al., 2003). We gathered a total of 200,000 patche of ize 32x32, randomly ampled from natural image from the dataet (van Hateren et al., 1998). The hitogram of their equivalent-michelon contrat i regarded a the prior ditribution of the environment (ee Figure 9c). The detailed decription of thi proce i dicued in Appendix C. In early viual perception ytem, contrat information i encoded by a population of neuron with contrat electivity in a oft-threholding manner. One traditional model characterize the neuron repone a a function of the contrat c via the Naka-Ruhton equation (Naka et al., 1966), h(c) = h max c q c q 50 + c q (36) where h max i the maximum poible firing rate, c 50 i the emi-aturation contrat o that h(c 50 ) = 0.5 h max and q i an exponent parameter characterizing the teepne of the curve near c 50. Uing our framework, we can predict the ditribution of emi-aturation contant c 50 within a population and compare that to phyiology data (Clatworthy et al., 2003) (ee Figure 9e). Our prediction ugget that the monkey V1 neuron are roughly performing Infomax (p 0.15) trategy while the cat triate cortex neuron are uing a larger value of p (p 0.75). A we can ee from Figure 9e, the fit for c 50 ditribution of cat triate cortex i wore than the fit for c 50 ditribution of monkey V1. The neural population in cat V1 eem to be adapted to maller contrat value. Thi may be due to the mimatch between the natural image dataet and the true viual environment of the animal. 17

18 a c homogeneou pace firing rate b meta-tuning curve d p=0.0 1% 10% 100% contrat p=0.5 1% 10% 100% contrat p=1.0 1% 10% 100% contrat prior ditribution image data mooth fit firing rate e 1% 10% 100% contrat f 1% 10% 100% contrat 1% 10% 100% contrat 1% 10% 100% contrat data likelihood p*=0.15 cat triate monkey V1 p*=0.75 neural denity cat triate monkey V1 predicted p 1% 10% 100% c 50 ditribution 1% 10% 100% c 50 ditribution 1% 10% 100% c 50 ditribution Figure 9: The analyi of optimal L p optimal neural population to encode contrat value in natural image. (a)-(d) cartoon example of L p -optimal neural population are derived baed on a homogeneou neural population and the optimal meta-tuning curve, which i determined by the prior ditribution of equivalent-michelon contrat extracted from natural image. The p value are 0, 0.5 and 1. (e)-(f) the predicted of c 50 ditribution for the entire population i compared with phyiology data reproduced from (Clatworthy et al., 2003) on cat triate cortex and monkey V1, which ugget p 0.15 for the monkey and p 0.75 for the cat. 6 Dicuion We have propoed a family of efficiency criteria for neural coding. Each efficiency criterion uniquely determine an optimal way of encoding a calar timulu with an arbitrary prior ditribution. The efficiency criteria are parameterized by a parameter p 0 aociated with the underlying goal of minimizing the L p recontruction error when uing a maximum likelihood decoder. Thee efficiency criteria naturally generalize everal pecial cae that have received much attention in the literature, e.g. the Infomax cae (p 0) or the minimal mean quared error (MMSE) cae (p = 2). For each optimality criterion and a timulu with known prior, we analytically derived the optimal tuning curve for a ingle neuron. To extend thi reult to determine optimal neural population, we propoed to ue the meta-tuning curve and howed that the optimal meta-tuning curve i identical to the optimal tuning curve for a ingle neuron with Gauian noie. Thee prediction baed upon different optimality criteria are teted againt previouly meaured characteritic of everal early viual ytem for different 18

19 animal. Prediction correponding to low value of p provide the bet match, which ugget that the optimality criterion i near Infomax for the neural repreentation being conidered. In our model and analyi, we have made the key aumption that the decoder i aymptotically unbiaed. Thi implie that the reult are trictly valid only in the low noie regime, e.g. when there i ufficient encoding time and/or a ufficient number of neuron. However, baed on numerical imulation we found that it i reaonably afe to relax the long encoding time aumption in particular if the neural population ize i large and/or the optimal criterion parameter p i mall. Many behavioral tudie alo ugget that human and other animal make deciion that are often biaed due to the effect of prior belief (Knill et al., 1996; Wei et al., 2015). With numerical imulation we howed that at hort encoding time, the Bayeian MAPE decoder i indeed performing better than the unbiaed MLE decoder, and lightly better than our analytic prediction. In fact, the performance of the MLE i lower bounded by our theoretical prediction (olid line in Figure 6) while the performance of the MAPE benefit from the prior information. Thu our reult are trictly valid only when auming an MLE decoder. In ection 2.2, we analyzed the Poion noie model and the contant Gauian noie model. Similar analyi can be applied to other noie model where the output variance depend upon the output mean. For neural population, we aumed that the output noie of an individual neuron i independent from the other, thu implifying the computation of the total Fiher information of the population. If the output noie ha a correlated tructure, then the total Fiher information i no longer the um of the Fiher information of individual neuron. Analyi of neural population decribed by a metatuning curve with correlated noie i a ubject for further invetigation. In concluion, we believe that our model how the utility of exploring different recontruction error criteria for analyzing neural repone in perceptual ytem. The parameter p decribe whether the neural ytem i adapted to more or le robut error tatitic, and we have obtained ome etimate of thi parameter from data on early viual proceing neuron in a number of different animal. It will be intereting to explore how the parameter p change a information propagate through variou tage of the perceptual ytem. We are alo currently invetigating how thi analyi can be extended to higher-dimenional timuli and to more complex noie model. Acknowledgment Thi work ha been upported by grant from the Office of Naval Reearch and Air Force Office of Scientific Reearch. We alo thank Xue-Xin Wei for fruitful dicuion and feedback on the manucript. A Fiher Information The concept of Fiher Information provide a tatitical characterization of how well a random variable r can be ued to etimate an underlying parameter under a tochatic 19

20 model p(r ). If a family of ditribution p(r ) i characterized by a one dimenional parameter, then the Fiher information i defined a (ee (Cover et al., 1991)), ( ) 2 d I() = log p(r ) d (37) A.1 Link to Popular Lo Function p(r ) Mutual Information Limit One poible meaurement of neural coding quality i the mutual information. Meauring mutual information doe not require an explicit etimator ŝ(r). Intead, it directly meaure the level of dependency between the neural repone r and the input timulu. The link between mutual information I mutual (r, ) and the Fiher information matrix wa etablihed in (Brunel et al., 1998). I mutual (r, ) = 1 2 log I() + cont. (38) Here we will not repeat the careful and delicate derivation but the main idea i baed on the fact that an efficient and unbiaed etimator ŝ i approximately ditributed a a Gauian with mean and variance I() 1. The conditional entropy of uch Gauian random variable i locally 1/2 log(i() 1 ) + cont and by averaging the local conditional entropy, we can get the mutual information. In term of Fiher information matrix, we want to maximize the right ide of Eq.(38). Cramer-Rao Lower Bound Another poible way to meaure coding quality i to ue the L 2 norm to meaure the error vector ŝ. Such L 2 norm i related to the Fiher information matrix via the Cramer-Rao lower bound (Cover et al., 1991). For any unbiaed etimator ŝ(r), e.g. the maximum likelihood etimator (MLE), Var[ŝ(r) ] I() 1 (39) A a lower bound, the Cramer-Rao bound can be attained by the MLE ŝ(r) due to i aymptotic efficiency (Cover et al., 1991). In order to calculate the mean L 2 error, one can find the attainable lower bound both locally at a given point or globally averaged over all, ŝ 2 r = Var[ŝ(r) ] I() 1. (40) ŝ 2 r, I() 1 (41) Compare thi with Eq.(38), we now derive another way of evaluating the Fiher information matrix. In order to minimize the mean L 2 error, one hould minimize the right ide of Eq.(41). For a more complete work regarding the relationhip between Fiher information and the Cramer-Rao lower bound, the reader i referred to (Pilarki et al., 1999) Aymptotic L p Limit 20

21 A natural generalization of L 2 metric to evaluate the difference ŝ i the L p metric for other value of p. In order to obtain the optimal L p population code, one can intead olve the optimization problem to minimize the mean L p norm of the difference ŝ by evaluating the p-th abolute moment of a Gauian random variable with zero mean and variance I() 1. Such family of optimization problem parameterized by p can provide a natural connection between two traditional optimal criteria the Infomax and MMSE (L 2 -min) ŝ(r) p r, cont(p) I() p/2 (42) When p = 2, it i clear that the right ide of Eq.(42) i the ame a the Cramer-Rao lower bound in Eq.(41) up to ome contant. In the limit of p 0, we can ue the replica trick to how that minimizing the right ide of Eq.(42) i equivalent to maximizing the mutual information term in Eq.(38). I() p/2 1 lim p 0 p = 1 log I() (43) 2 Thee characterization of lo function in Eq.(38), Eq.(41) and Eq.(42) by uing Fiher Information implifie the proce of finding the optimal neural code. A.2 Fiher Information Example In order to apply the concept of Fiher Information to analyze the performance of neural code, here we calculate the Fiher Information for a ingle neuron with Poion noie model or contant Gauian noie model. Poion Spiking Model The firt model i the Poion piking model. If the neuron elicit a random number of pike r during a given time window T i a Poion random variable with rate T h() P (r = N ) = 1 N! ( T h())n exp( T h()) (44) log P (r = N ) = log(n!) + N log ( T h()) T h() (45) ( ) d N dx log P (r = N ) = h () h() T (46) For Poion random variable N with rate T h() we know that, N = T h(), N 2 = T h() + ( T h()) 2 (47) Uing thi reult we know ( ) 2 d I() = log P (r = N ) d ( ) 2 N = h () 2 h() T = T h () 2 h() (48) 21

22 If the optimal Fiher Information I() i known, the optimal nonlinearity h() can be derived by olving the above ordinary differential equation ( h() I(ξ) dξ ) 2. (49) Contant Gauian Noie Model In the econd model, we aume the random number of pike can be any real number. The additive noie in each unit time window i σ0 2 therefore the total number of pike r k that ha been oberved over a time window of length T i a Gauian random variable with mean T h() and variance σ0 T 2 ( 1 p(r ) = 2πσ 2 0 T exp 1 ) (r T 2σ0 T 2 h())2 (50) log p(r ) = log(2πσ2 0 T ) 2σ0 T (r T 2 h())2 (51) d d log p(r ) = h () (r T h()) (52) σ 2 0 Uing thi reult we know that the Fiher information for a neuron with contant Gauian noie i ( ) 2 d I() = log p(r ) d = h () 2 (r T h()) 2 = T h () 2 (53) σ0 4 σ0 2 If the optimal Fiher Information I() i known, the optimal nonlinearity h() can be derived by olving the above ordinary differential equation h() I(ξ) dξ. (54) In Eq.(48) and Eq.(53) we have derived the Fiher Information of a ingle neuron with Poion or contant Gauian noie model. In order to generalize from a ingle neuron to a population of neuron, we need the following reult to meaure the overall goodne of a population code. A.3 Fiher Information for Neuron with Independent Noie When each neuron in the population ha independent noie, here we prove that the total Fiher information of the population i the linear um of the Fiher information contributed by each individual neuron m p(r ) = p(r k ) I total () = k=1 m I k () (55) k=1 22

23 Uing the definition of Fiher Information, we know ( ) 2 d I total () = log p(r ) d ( m ) 2 d = d log p(r k ) k=1 (56) (57) When k l, we know the neural repone r k, r l are independent conditioned on. Therefore d d log p(r k ) d d log p(r l ) d = d log p(r k ) d d log p(r l ) = 0 (58) which i becaue d d log p(r k ) = d p(r d k ) p(r k )dr k = d ( p(r k ) d p(r k ) dr k ) = 0. (59) A a concluion, the total Fiher Information for a population of neuron with independent Poion/contant Gauian noie i equal to the linear um of the Fiher information of each neuron. B Etimating the Ditribution over Local Orientation We extracted orientation tatitic for natural image from a tandard image databae (van Hateren et al., 1998). Firt we randomly ampled 200,000 quare patche (16pixby-16pix) acro the entire databae. We then created a et of inewave grating filter with a fixed patial frequency that wa cloe to the human peak enitivity (approximately 4 cycle per viual degree or 8 pixel/cycle) but variou phae and 360 different orientation (0 to with 0.5 pacing). The dominant orientation of each patch wa determined by the maximum repone acro all thee filter. To mitigate the effect of pixel-wie noie or quantization effect, we only ued thoe patche with high filter repone level (top 50%). The reulting prior ditribution i very imilar to previouly meaured ditribution (e.g. Girhick et al. (2011)) and i hown in Figure 8c. We ued a pline function to fit the cumulative of the empirical hitogram in order to obtain a mooth verion of the denity f(θ). C Equivalent-Michelon Contrat Originally, the Michelon contrat i defined for inuoid grating baed on it max/min luminance c = L max L min L max + L min (60) 23

24 It i clear the the Michelon contrat ha a value between 0 and 1. For any patche of non-inuoid grating, we determine it equivalent-michelon contrat in the following way. For each image patch, we ue a et of 64 odd-gabor filter g gabor (x, y) of different orientation θ and wavelength λ to convolute with natural image patche to obtain local repone. Specifically, the Gabor filter are g gabor (x, y) = g normal (x, y) g inuoid (x, y) (61) ) ) g normal (x, y) = exp ( x 2 + y 2, g 2σ 2 inuoid (x, y) = in (2π x (62) λ x = x co θ + y in θ, y = x in θ + y co θ, σ = 1 ln 2 2 b + 1 π 2 2 b 1 λ (63) where the orientation θ take 8 value uniformly ampled from the range [0, π], the wavelength λ take 8 value uniformly ampled in the logarithm pace from 4 to 85.3 pixel per cycle. The ize of Gauian filter σ i automatically determined by the wavelength λ and a fixed octave value b = 1.5 in order to bet match the propertie of imple cell in the primary viual cortex. With uch a filter bank of 64 Gabor filter, we calculate the equivalent-michelon contrat for each image patche. For each Gabor filter, we ue the correponding Gauian filter g normal (x, y) to compute the local mean luminance to model luminance adaptation. We alo ue the correponding inuoid filter g inuoid (x, y) to contruct a teting inuoid grating L ave + L amp g inuoid (x, y). By properly chooing the parameter L ave and L amp, we can match both the Gabor-filter repone and the Gauian-filter repone. The equivalent-michelon contrat i then determined by the Michelon contrat of thi teting grating: L max = L ave + L amp, L min = L ave L amp c = L amp L ave (64) The above proce i ummarized in Figure 10. The local contrat value of each image patche i then determined by taking the maximum among the 64 equivalent-michelon contrat value calculated uing the Gabor filter bank. Thi max operation i taken in order to match the normalization computation taken place in the viual perception pathway (Carandini et al., 2012). Neuron that are reponding to a low contrat value often appear to be ilent (normalized out) when there i a neighbor neuron reponding to a ignificantly larger contrat. 24