A Radically New Theory of how the Brain Represents and Computes with Probabilities

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1 A Radically New Theory of how the Brain Represents and Comptes with Probabilities Rod Rinks, Nerithmic Systems, 468 Waltham St., Newton, MA 2465 SA & Volen Center for Complex Systems, Brandeis niversity, 45 Soth St., Waltham, MA 2453 SA ORCID ID: X Abstract The brain is believed to implement probabilistic reasoning and to represent information via poplation, or distribted, coding. Most previos probabilistic poplation coding (PPC) theories share basic properties: ) continos-valed nits; 2) flly/densely-distribted codes; 3) graded synapses; 4) rate coding; 5) nits have innate nimodal tning fnctions (TFs); 6) nits are intrinsically noisy; and 7) noise/correlation is generally considered harmfl. We present a radically different theory that assmes: ) binary nits; 2) only a small sbset of nits, i.e., a sparse distribted representation (SDR) (a.k.a. cell assembly, ensemble), comprises any individal code; 3) fnctionally binary synapses; 4) signaling formally reqires only single (i.e., first) spikes; 5) nits initially have completely flat TFs (all weights zero); 6) nits are far less intrinsically noisy than traditionally thoght; rather 7) noise is a resorce generated/sed to case similar inpts to map to similar codes, controlling a tradeoff between storage capacity and embedding the inpt space statistics in the pattern of intersections over stored codes, epiphenomenally determining correlation patterns across nerons. The theory, Sparsey, was introdced 2+ years ago as a canonical cortical circit/algorithm model achieving efficient spatiotemporal pattern learning/recognition, bt was not elaborated as an alternative to PPC-type theories. Here, we show that: a) the active SDR simltaneosly represents both the most similar/likely inpt and the entire (coarsely-ranked) similarity/likelihood distribtion over all stored inpts (hypotheses); and b) given an inpt, Sparsey s code selection algorithm, which nderlies both learning and inference, pdates both the most likely hypothesis and the entire likelihood distribtion (cf. belief pdate) with a nmber of steps that remains constant as the nmber of stored items increases. Keywords Sparse distribted representations, probabilistic poplation coding, cell assemblies, seqence learning and recognition, neral noise, canonical cortical circit/algorithm Acknowledgement I d like to thank the people who have encoraged me to prse this theory over the years, inclding Dan Bllock, Jeff Hawkins, John Lisman, Josh Alspector, Tom McKenna, Andrew Browning, and Dan Hammerstrom. I d also like to thank my colleages at Nerithmic Systems, Greg Lesher, Jasmin Leveille, Oliver Layton, Harald Rda, and Nick Nowak for their help developing these ideas. This work has been partially spported by DARPA contracts FA865-3-C-7432 and N73-9-C-238, ONR contract N4-2-C-539, and NIH Training grant, 5 T32 NS7292.

2 Introdction It is widely acknowledged that the brain mst implement some form of probabilistic reasoning to deal with ncertainty in the world (Poget, Beck et al. 23). However, exactly how the brain represents probabilities/likelihoods remains nknown (Ma and Jazayeri 24, Pitkow and angelaki 26). It is also widely agreed that the brain represents information with some form of distribted a.k.a. poplation, cellassembly, ensemble code [see (Barth and Polet 22) for relevant review]. Several poplation-based probabilistic coding theories (PPC) have been pt forth in recent decades, inclding those in which the state of all nerons comprising the poplation, i.e., the poplation code, is viewed as representing: a) the single most likely/probable inpt vale/featre (Georgopolos, Kalaska et al. 982); or b) the entire probability/likelihood distribtion over featres (Zemel, Dayan et al. 998, Poget, Dayan et al. 2, Poget, Dayan et al. 23, Jazayeri and Movshon 26, Ma, Beck et al. 26, Boerlin and Denève 2). Despite their differences, these approaches share fndamental properties, a notable exception being the spike-based model of (Boerlin and Denève 2).. Neral activation is continos (graded). 2. All nerons in the coding field formally participate in the active code whether it represents a single hypothesis or a distribtion over all hypotheses. Sch a representation is referred to as a flly distribted representation. 3. Synapse strength is continos (graded). 4. These approaches have generally been formlated in terms of rate-coding (Sanger 23), which reqires significant time, e.g., order tens of ms, for reliable decoding. 5. They assme a priori that tning fnctions (TFs) of the nerons are nimodal, bell-shaped over any one dimension, and conseqently do not explain how sch TFs might develop throgh learning. 6. Individal nerons are assmed to be intrinsically noisy, e.g., firing with Poisson variability. 7. Noise and correlation are viewed primarily as problems that have to be dealt with, e.g., redcing noise correlation by averaging. At a deeper level, it is clear that despite being framed as poplation models, they are really based on an nderlying localist interpretation, specifically, that an individal neron s firing rate can be taken as a perhaps noisy estimate of the probability that a single preferred featre (or preferred vale of a featre) is present in its receptive field (Barlow 972). While these models entail some method of combining the otpts of individal nerons, e.g., averaging, each neron is viewed as providing its own individal, i.e., localist, estimate of the inpt featre, i.e., each neron possesses its own independent (typically bell-shaped) TF. For example, this can be seen qite clearly in Fig. of (Jazayeri and Movshon 26) wherein the first layer cells (sensory nerons) are nimodal and therefore can be viewed as detectors of the vale at their modes (preferred stimls) and the pooling cells are also in -to- correspondence with directions. This nderlying localist interpretation is present in the other PPC models referenced above as well. However, there are several compelling argments against sch localistically-rooted conceptions. From an experimental standpoint, a growing body of research sggests that individal cell TFs are far more heterogeneos than classically conceived (Yen, Baker et al. 27, Cox and DiCarlo 28, Smith and Hässer 2, Bonin, Histed et al. 2, Mante, Sssillo et al. 23, Nandy, Sharpee et al. 23, Nandy, Mitchell et al. 26), also described as having mixed selectivity (Fsi, Miller et al. 26). And, the greater the fidelity with which the heterogeneity of TFs is modeled, the less neronal response variation that needs to be attribted to noise, leading some to qestion the appropriateness of the traditional concept of a single neron I/O fnction as an invariant TF pls noise (Deneve and Chalk 26). From a formal standpoint, the limitation has long been pointed ot that the maximm nmber of featres/concepts, e.g., oriented edges, directions of movement, that can be stored in a localist coding field of N nits is N. More importantly, as or own work has shown, the comptational time efficiency with which featres/concepts 2

3 can be stored and retrieved, is far greater for memories in which items are represented with sparse distribted representations (SDRs) than for localist memories (Rinks 996, Rinks 2, Rinks 24). The theory described herein, Sparsey, constittes a radically new way of representing and compting with probabilities, diverging from most existing PPC theories in many fndamental ways, inclding:. The nerons comprising the coding field need only be binary. 2. Individal represented items / hypotheses are represented by fixed-size, sparsely chosen sbsets of the poplation, i.e., SDRs (Rinks 996, Rinks 24). 3. Decoding (read-ot) of the most likely hypothesis and of the entire distribtion by downstream comptations, reqires only binary synapses. 4. Signaling can be commnicated via a wave of contemporaneosly arriving first-spikes (e.g., within a few ms window, likely organized by local gamma) from an afferent SDR code to a downstream comptation (inclding recrrently to the sorce coding field) and is ths in principal, -2 orders of magnitde faster than rate coding. This means Sparsey is not formally a spiking model. 5. The initial weights of all afferent synapses to all nerons comprising the field are zero, i.e., the TFs are completely flat. Roghly nimodal TFs [as wold be revealed by low-complexity probes, e.g., oriented bars spanning a cell s receptive field (RF)] emerge as a side-effect of the model s single/few-trial learning process of laying down SDRs in sperposition provided that the process of choosing those SDRs preserves similarity (which Sparsey s learning algorithm does). 6. Nerons are far less inherently noisy than has been classically spposed. Most observed noise has likely been de to experimental limitations, i.e., the inability to trly closely control inpt conditions. At an algorithmic, or information-bearing level, individal nerons are simply on or off on any given comptational cycle. We make no se of precise spike times, bt assme that some meta-circitry organizes (temporally collects) transmission of en masse synaptic signals from an SDR coding field within some small window of arrival at a downstream decoding field. Or working assmption is that this is the prpose of the local gamma cycle (Fries 29, Bzsáki 2, Igarashi, L et al. 24, Watros, Fell et al. 25). 7. While nerons are inherently less noisy than has been assmed, the canonical, mesoscale (i.e., the cell assembly scale) circit does explicitly se noise as a resorce. That is, noise (presmably mediated by neromodlators, e.g., NE, ACh) is explicitly generated and injected into the code selection process to achieve a specific coding goal. That goal is to make the SDR code that is activated in response to an inpt be increasingly random as a fnction of novelty, i.e., to minimize the intersection of the activated code with all previosly stored codes. More generally, the goal is that the overall set of codes stored in a coding field has the property that similar inpts map to similar codes ( SISC ), where the similarity measre for SDR codes is size of intersection. This has straightforward implications on correlation, described below. SDR admits a completely different concept for representing and compting with probabilities than is possible nder either localist or flly distribted representations. Instead of representing an inpt/featre, X, by the state of a single neron as in localism, or by a vector of real vales over all nerons comprising the coding field, as in flly distribted coding (e.g., as in a hidden level of a mlti-layer perceptron model), in SDR, X is represented by a sbset of nerons, specifically, a small sbset of the whole coding field, which we refer to as X s code. All nerons participating in the code are flly active (in a binary sense) and the rest of the coding field s nerons are completely off, which is closely consistent with the combinatorial coding framework described and analyzed in (Osborne, Palmer et al. 28). Ths: A. Gradations in the certainty, or probability, that X is present in the receptive field (RF) of the SDR coding field can be represented by different fractions of X s code being active (see Fig. 2). 3

4 B. Commnication of X s probability to target (downstream, or sbseqent) comptations i.e., decoding can be achieved by nerons participating in sch target comptations simply smming their binary synaptic inpts from the (sorce) field in which X is active. Ths, representing and sing graded vales, e.g., probabilities, reqires only binary nerons and binary synapses. There is no need for explicit localist representation of graded vales, anywhere in the system/comptation. More specifically, there is no need to represent probabilities/likelihoods localistically in the form of firing rates (nor in the form of exact spike times within some window) and there is no need to represent conditional probabilities, or strengths of association, via continos/graded weights. Moreover, althogh nerons presmably commnicate primarily via spikes, conceiving the signal sent from an active SDR code as a vector of contemporaneosly arriving first spikes (to a target field) essentially removes the need to cast or treatment in terms of a spiking model with its assmed (typically Poisson) intrinsic noisiness. Rather, or approach has more the character of an algorithm operating according to a discrete clock (whose basic cycle, we sppose is a gamma cycle). Ths, or approach does not assme intrinsic noise that has to be dealt with, bt rather, as we ll describe, injects a state-dependent amont of noise by maniplating nerons transfer fnctions, specifically, their nonlinearities, to achieve certain coding goals, as discssed below. This nderscores a strongly distingishing property of Sparsey. The nonlinearity of the principal cells is not static as is tre of most models, bt highly dynamic: specifically, the nonlinearities of all principal cells comprising a coding field are modlated en masse, in correlated fashion, and on a fast time scale, e.g., ~ ms, as a fnction of a global (to the coding field) measre of the familiarity (inverse novelty) of the total inpt to the coding field. The niqe property of SDR that the similarity of represented items/featres can be represented by the intersection size of their codes, i.e., SISC, has been pointed ot in the past (Palm, Schwenker et al. 995, Rinks 996, Rachkovskij and Kssl 2, Kanerva 29). However, the possibility of: a) interpreting the fraction of a featre s SDR code that is active as the probability/likelihood of that featre (hypothesis), and b) interpreting the code active in an SDR coding field, whether it be a reactivation of a previosly stored code or a novel code (which nevertheless, may generally overlap with many previosly stored codes), as a distribtion over all stored codes (and ths, over all represented inpts/featres) stored in the field was introdced in (Rinks 22). Similarity preservation, involving a continos measre, e.g., Eclidean distance, has also been established for flly distribted coding models, e.g., (Bogacz 27). However, to or knowledge, sch codes have not been interpreted as simltaneosly representing both the most likely hypothesis and the entire distribtion. In any case, as noted above, in sch models, decoding by downstream comptations reqires either graded synapses or rate-coding. Above, we refer to the RF of the SDR coding field rather than to the RF of an individal neron. This is becase we reqire that, to first approximation, all nerons in the SDR coding field have the same RF in order to assert that the codes in that field represent featres present in the RF. This constraint can be relaxed to some extent, bt it facilitates initial exposition and analysis. In any case, this constraint actally serves to frther nderscore the massive glf between Sparsey and previos PPC theories for which this constraint is not needed, even approximately, de to their nderlying localist conception. Sparsey evinces a new paradigm of neroscience in which the ensemble (SDR, cell assembly), not the single neron, is considered the fndamental fnctional nit, i.e. a move away from the Neron Doctrine as for example, advocated in (Yste 25, Fsi, Miller et al. 26, Schneidman 26). We expect that with rapidly advancing methods [recent reviews: (Hamel, Grewe et al. 25, Jercog, Rogerson et al. 26)] allowing observation of the activities of all nerons in sbstantial (e.g., order hypercolmn/barrel-sized) volmes and with progressively greater temporal precision, isses sch as or connectivity assmption as well as the dynamics implied in or core algorithm will be addressable. In the reslts section, we present two simlation-backed examples showing the mechanistic details of how Sparsey s core algorithm, the Code Selection Algorithm (CSA) (Rinks 996, Rinks 24, Rinks 4

5 and Lisman 25, Rinks 28, Rinks 2, Rinks 23, Rinks 24), activates not only the code of the best-matching, or most likely, inpt, bt also activates the entire similarity/likelihood distribtion, with coarsely-ranked fidelity, over all stored inpts. Moreover, the CSA accomplishes this with a nmber of steps that remains constant as the nmber of stored inpts increases, a property that we call fixed time complexity. The first example shows this for the case of prely spatial inpts. The second example shows that when the coding field is recrrently connected, the entire distribtion is pdated from time T to T+ [which can be viewed as belief pdate (Pearl 988)] in a way that is consistent with the statistics of the domain, again, in fixed time. Moreover, as explained previosly (Rinks 2, Rinks 24), the CSA also stores, or learns, new inpts, spatial or spatiotemporal, in fixed time. In fact, Sparsey/CSA can be viewed as an adaptive hashing method which learns a locality-sensitive, i.e., similarity-preserving, hash fnction from the data (which can be spatial or spatiotemporal). While other nerally inspired hashing models have fixed-time best-match retrieval (Salakhtdinov and Hinton 27, Salakhtdinov and Hinton 29, Graman and Fergs 23), they do not also have fixed-time learning. In fact, none of the crrently stateof-art hashing models described in a recent review (Wang, Li et al. 26) possess both fixed-time learning and fixed-time best-match retrieval. Althogh time complexity considerations like these have generally not been discssed in the probabilistic poplation coding literatre, they are essential for evalating the overall plasibility of models of biological cognition, for while it is ncontentios that the brain comptes probabilistically, we also need to explain the extreme speed with which these comptations, which are over potentially qite large hypothesis spaces, occr. A crcial reason why Sparsey achieves fixed time performance for both learning and best-match retrieval is its niqe and simple method for compting the global familiarity, G, of an inpt and sing it to control the selection of a code for that inpt. By global, we mean that G is a fnction of all cells comprising the coding field, in contrast to local familiarity, V i, which is cell i s local (depending only on its synaptic inpts) measre of its match to the inpt. Crcially, compting G does not reqire explicitly comparing the new inpt to every stored inpt (nor to a log nmber of the stored inpts as is the case for tree-based methods). Rather, the G comptation, whose time complexity is dominated by a single pass over the fixed nmber of afferent weights to the field, implicitly performs these comparisons simltaneosly, in an algorithmically parallel fashion. Algorithmic parallelism means that single atomic operations affect mltiple represented (stored) items. Ths, operationally, algorithmic parallelism is very close if not identical to distribted representation : yo cannot have one withot the other. We emphasize that algorithmic parallelism and machine parallelism are orthogonal resorces and are flly compatible. A simplified version of the CSA, sfficient for this paper s examples, is given in Table, bt we briefly smmarize it here. CSA Step comptes the inpt sms for all Q K cells comprising the coding field. Specifically, for each cell, a separate sm is compted for each of its major afferent synaptic projections, e.g., bottom-p (), horizontal (H), and top-down (D) projections, the latter two of which provide recrrence to the coding field. This is the step reqiring the aforementioned single pass over the field s afferent weights. In Step 2, these sms are normalized and in Step 3, the normalized sms are (optionally nonlinearly transformed and) mltiplied, yielding the V vales. In Steps 4 and 5, G is compted as the average max V across the Q CMs. In the remaining CSA steps, G is sed, in each CM, to nonlinearly transform the V distribtion over the K cells into a final ρ distribtion from which a winner is picked. G s inflence on the distribtions can be smmarized as follows. a) When high global familiarity is detected (G ), those distribtions are exaggerated to bias the choice in favor of cells that have high inpt smmations, and ths, high local familiarities, V i, which acts to increase correlation. b) When low global familiarity is detected (G ), those distribtions are flattened so as to redce bias de to local familiarity, which acts to increase the expected Hamming distance between the selected code and previosly stored codes, i.e., to decrease correlation. Since the V vales represent signal, exaggerating the V distribtion in a CM increases signal whereas flattening it increases noise. The above behavior (and its smooth interpolation over the range, G= to G=) 5

6 is the means by which Sparsey achieves SISC. And, it is the enforcement (statistically) of SISC dring learning, which ltimately makes possible the immediate (fixed time complexity) retrieval of the bestmatching (or most likely, most relevant) hypothesis and the simltaneos fixed-time pdate of all stored hypotheses (i.e., via algorithmic parallelism, not machine parallelism) with each new inpt. Novel explanation of noise and correlation In recent years, there has been mch discssion abot the natre, cases, and ses, of correlations and noise in cortical activity; see (Cohen and Kohn 2, Kohn, Coen-Cagli et al. 26, Schneidman 26) for reviews. Most investigations of neral correlation and noise, especially in the context of PPC theories, assme a priori: a) fndamentally noisy nerons, e.g., Poisson spiking; and b) tning fnctions (TFs) of some general form, e.g., nimodal, bell-shaped, and then describe how noise/correlation affects the coding accracy of poplations with sch TFs (Abbott and Dayan 999, Moreno-Bote, Beck et al. 24, Franke, Fiscella et al. 26, Rosenbam, Smith et al. 27). Specifically, these treatments measre correlation in terms of either mean spiking rates ( signal correlation ) or spikes themselves ( noise correlations ). However, as noted above, or theory makes neither assmption. Rather, in or theory, noise (randomness) is actively injected implemented via the G-dependent modlation of the neronal transfer fnction dring learning to achieve the goals described above. Ths, the pattern of correlations amongst nits (nerons) simply emerges as a side-effect of cells being selected to participate in codes. The overarching goal of the learning process is simply to enforce SISC. However, enforcing SISC in the context of an SDR coding field realizes a balance between: a) maximizing the storage capacity of the coding field, and b) embedding the similarity strctre of the inpt space in the set of stored codes, which in trn enables fixed-time best-match retrieval. Interestingly, in exploring the implications of shifting focs from information theory to coding theory in terms of inflence pon theoretical neroscience, (Crto, Itskov et al. 23) have pointed to this same tradeoff, thogh their treatment ses error rate (coding accracy) instead of storage capacity. We point ot that nderstanding how neral correlation ltimately affects things like storage capacity is considered largely nknown and an active area of research (Latham 27). Or approach implies a straightforward answer. Minimizing correlation, i.e., maximizing average Hamming distance over the set of codes stored in an SDR coding field, maximizes storage capacity. Increases of any correlations of pairs, triples, or sbsets of any order, of the coding field s cells decreases capacity. 6

7 2 Reslts A single SDR code represents an entire similarity/probability/likelihood distribtion: Conceptal introdction Fig. a shows Sparsey s particlar SDR format. The coding field consists of Q winner-take-all (WTA) competitive modles (CMs), each consisting of K binary nerons. Here, Q=7 and K=7. Ths, all codes have exactly Q active nerons and there are K Q possible codes. We assme that the inpt field of binary nerons (hexagons), e.g., representing an 8x8 pixel patch of visal field, is completely connected to the coding field (ble lines represent weights), i.e., the aforementioned RF of SDR coding field. All weights are initially zero. Fig. b shows a particlar inpt A, which has been associated with a particlar code, φ(a); here, ble lines indicate the bndle [cf. Synapsemble, (Bzsáki 2)] of weights that wold be increased from to to store this association (memory trace). SDR Coding Field ( mac ) φ( A) Inpt Field A a) Fig. a) The sparse distribted representation (SDR) coding field (macrocolmn, mac ) consists of Q=7 WTA competitive modles. An inpt field of binary nerons is completely connected to the coding field. b) A particlar inpt A, its code, φ(a), consisting of Q active (black) nerons, and the bndle of weights that wold be increased to form the association are shown Fig. 2 shows how the strength of presence, i.e., (posterior) probability, of a featre in the coding field s RF is represented in or SDR-based theory. The figre shows five hypothetical inpts, A-E, which have been learned, i.e., associated with codes, φ(a) - φ(e). We hand-chose these particlar codes to be consistent with the principle that similar inpts shold map to similar codes (SISC). That is, inpts B to E have progressively smaller overlaps with A and therefore codes φ(b) to φ(e) have progressively smaller intersections with φ(a). The CSA has been shown to statistically enforce SISC for both the spatial and spatiotemporal (seqential) inpt domains (Rinks 996, Rinks 28, Rinks 2, Rinks 23, Rinks 24). For inpt spaces for which it is plasible to assme that inpt similarity correlates with probability/likelihood, the single active code can therefore also be viewed as a probability/likelihood distribtion over all stored codes. This is shown in the lower part of the figre. The leftmost panel at the bottom of Fig. 2 shows that when φ(a) is % active, the other codes are partially active in proportions that reflect the similarities of their corresponding inpts to A, and ths the probabilities/likelihoods of the inpts they represent. The remaining for panels show inpt similarity (probability/likelihood) approximately correlating with code overlap. b) 7

8 Inpt Item with Item A Code Name Code with φ(a) A φ( A) φ( A) φ( A) = 7 B φ(b) φ(b) φ( A) = 4 C φ(c) φ(c) φ( A) = 3 D φ(d) φ(d) φ( A) = 2 E φ(e) φ(e) φ( A) = (%) with φ(a) with φ(b) with φ(c) with φ(d) with φ(e) φ(a) φ(b) φ(c) φ(d) φ(e) φ(a) φ(b) φ(c) φ(d) φ(e) φ(a) φ(b) φ(c) φ(d) φ(e) φ(a) φ(b) φ(c) φ(d) φ(e) φ(a) φ(b) φ(c) φ(d) φ(e) Fig. 2 Illstration of how the probability/likelihood of a featre can be represented by the fraction of its code that is active. When φ(a) is flly active, the hypothesis that featre A is present can be considered maximally probable. Becase the similarities of the other featres to the most probable featre, A, correlate with their codes overlaps with φ(a), the probabilities/likelihoods of those featres are represented by the fraction of their codes that are active. In the intersection ( ) colmns, black indicates nits intersecting with either the inpt pattern A or its code, φ(a); gray indicates non-intersecting nits The example of Fig. 2 was constrcted to illstrate the desired property that the similarities, and ths, likelihoods, of all stored hypotheses are simltaneosly physically active whenever any single hypothesis is flly active. The next section demonstrates that the CSA achieves this property for prely spatial inpts and the following section, for the spatiotemporal case. Table presents a simplified version of the code selection algorithm (CSA) with the minimal steps needed for these demonstrations. Specifically, the simplifications, relative to (Rinks 24) are: a) we se a model with only one internal level, ths there are no D signals; b) all spatial inpts have exactly 2 active pixels, ths the normalizer can be constant, π = 2 ; and c) the internal level consists of a single coding field (mac), ths the H normalizer can be constant, π = Q. H 8

9 Table Simplified Code Selection Algorithm (CSA) 2 3 j RF Eqation i () = x( jw ) ( ji,) hi ( ) = x( jt, ) w( ji, ) j RF H i () = i () π w Hi () = hi () π H w max max λh λ Hi () i () t Vi () = λ i () t= 4 V ˆ max { Vi ( ) } 5 Short Description Compte raw (and H, if applicable) inpt sms. Compte normalized (and H, if applicable) inpt sms. In this paper s simlations, π = 2 and π = Q. H Compte local evidential spport for each cell. In this paper, λ = λ =, nless otherwise stated. j = i C Find the max V, ˆj j V, in each CM, C j Q G V ˆ q= k = Q Compte G as average ˆ γ + G G 6 η = + χ K G ( η ) σ = 4 ( ). e σ σ 2 3 σ ( η ) () i = e σ σ σ + + ( ) 2( V() i 3 ) 4 σ H V vale over the Q CMs Determine expansivity (η) of V-to- sigmoid fnction. In this paper, γ=2, χ=, G =. Sets σ so that the overall sigmoid shape is preserved over fll η range. σ 2 = 7., σ 3 =.4, σ 4 = 9.5 To each cell, apply sigmoid fnction, which collapses to ) constant fn, () i =, when G G () i ρ() i = In each CM, normalize relative () to final (ρ) ( k) probabilities of winning k CM Select a final winner in each CM according to the ρ distribtion in that CM, i.e., soft max. Single SDR code represents entire probability/likelihood distribtion: Spatial case Fig. 3a shows six inpts, Ito I 6 (which are disjoint for simplicity of exposition) that have been previosly stored in the model instance depicted in Fig. 3b. The model has a 2x2 binary pixel inpt level that is completely connected to all cells comprising the mac. The mac consists of Q=24 WTA CMs, each having K=8 binary cells. The second row of Fig. 3a shows a novel test stimls, I 7, and its varying overlaps (yellow pixels) with I to I 6. Given that all inpts are constrained to have exactly 2 active pixels, we can measre spatial similarity, sim ( I, I ), simply as size of intersection divided by 2 (shown as decimals nder inpts): x y 9

10 sim( I, I ) = I I 2 x y x y Fig. 3b shows the code, φ(i 7), activated in response to I 7, which by constrction is most similar to I. Black coding cells are cells that also won for I, red indicates active cells that did not win for I, and green indicates inactive cells that did win for I. The red and green cells in a given CM can be viewed as a sbstittion error. The intention of the red color for coding cells is that if this is a retrieval trial in which the model is being asked to retrn the closest matching stored inpt, I, then the red cells can be considered errors. Note however that these are sb-symbolic scale errors, not errors at the scale of whole inpts (hypotheses, symbols), as whole inpts are collectively represented by the entire SDR code. In this example appropriate threshold settings in downstream/decoding nits, wold allow the model as a whole retrn the correct answer given that 8 ot of 24 cells of I s code, φ(i ), are activated, similar to thresholding schemes in other associative memory models (Marr 969, Willshaw, Bneman et al. 969). Note however that if this was a learning trial, then the red cells wold not be considered errors: this wold simply be a new code, φ(i 7), being assigned to represent a novel inpt, I 7, and in a way that respects similarity in the inpt space. Fig. 3d shows the first key message of the figre. The active fractions of the codes, φ(i ) to φ(i 6), representing the six stored inpts, Ito I 6, are highly rank-correlated with the pixel-wise similarities of these inpts to I 7. Ths, the ble bar in Fig. 3d represents the fact that the code, φ(i ), for the best matching stored inpt, I, has the highest active code fraction, 75% (8 ot 24, the black cells in Fig. 3b) cells of φ(i ) are active in φ(i 7). The cyan bar for the next closest matching stored inpt, I 2, indicates that 2 ot of 24 of the cells of φ(i 2) (code note shown) are active in φ(i 7). In general, many of these 2 may be common to the 8 cells in{ φ( I7) φ( I) }. And so on for the other stored hypotheses. (The actal codes, φ(i ) to φ(i 6), are not shown; only the intersection sizes with φ(i 7) matter and those are indicated along right margin of chart in Fig. 3d.) We note that even the code for I 6 which has zero intersection with I 7 has two cells in common with φ(i ). In general, the expected code intersection for the zero inpt intersection condition is not zero, bt chance, since in that case, the winners are chosen from the niform distribtion in each CM: ths, the expected intersection in that case is jst Q/K. As noted earlier, we assme that the similarity of a stored inpt I X to the crrent inpt can be taken as a measre of I X s probability/likelihood. And, since all codes are of size Q, we can divide code intersection size by Q, yielding a measre normalized to [,]: LI ( ) = φ( I) φ( I) / Q 7 We also assme that I to I 6 each occrred exactly once dring training and ths, that the prior over hypotheses is flat. In this case the posterior and likelihood are proportional to each other, ths, the likelihoods in Fig. 3d can also be viewed as nnormalized posterior probabilities of the hypotheses corresponding to the six stored codes.

11 a) Learned Inpts φ(i 7 ) I 7 (Test Stim.) I I 2 I 3 I 4 I 5 I (yellow) of I 7 with I to I 6 (and as decimals) b) I 7 c) CM ρ V= d) L Likelihood (L) I I 2 I 3 I 4 I 5 I e) CM 7 f) V Fig. 3 In response to an inpt, the codes for learned (stored) inpts, i.e., hypotheses, are activated with strength that is correlated with the similarity (pixel overlap) of the crrent inpt and the learned inpt. Test inpt I7 is most similar to learned inpt I, shown by the intersections (yellow pixels) in panel a. Ths, the code with the largest fraction of active cells is φ(i ) (8/24=75%) (ble bar in panel d). The other codes are active in rogh proportion to the similarities of I7 and their associated inpts (cyan bars). (c) Raw () and normalized () inpt smmations to all cells in all CMs. V vales, which eqal the vales in this prely spatial case, are transformed to n-normalized win probabilities () in each CM via sigmoid transform whose properties, e.g., max vale of 255.3, depend on G and other parameters. vales are normalized to tre probabilities (ρ) and one winner is chosen in each CM (indicated in row of triangles: black: winner for I 7 that also won for I ; red: winner for I7 that did not win I : green: winner for I that did not win for I 7. (e, f) Details for CMs, 7 and 5. Vales in second row of V axis are indexes of cells having the V vales above them. Some CMs have a single cell with mch higher V and ltimately ρ vale than the rest (e.g., CM 5), some others have two cells that are tied for the max (e.g., CMs 3, 9, 22) We acknowledge that the likelihoods in Fig. 3d may seem high. After all, I7 has less than half its pixels in common with I, etc. Given these particlar inpt patters, is it really reasonable to consider I to have sch high likelihood? Bear in mind that or example assmes that the only experience this model has of the world are single instances of the six inpts shown. We assme no prior knowledge of any nderlying statistical strctre generating the inpts. Ths, it is really only the relative vales that matter and we cold pick other parameters, notably in CSA Steps 6-8, that wold reslt in a mch less expansive sigmoid nonlinearity, which wold reslt in lower expected intersections of φ(i 7) with the learned codes, and ths lower likelihoods. The main point is simply that the expected code intersections correlate with inpt similarity, and ths, likelihood.. V CM 5

12 Fig. 3c shows the second key message: the likelihood-correlated pattern of activation levels of the codes (hypotheses) apparent in Fig. 3d is achieved via independent soft max choices in each of the Q CMs. Fig. 3c shows, for all 96 mac cells, the traces of the relevant variables sed to determine φ(i 7). The raw inpt smmation from active pixels is indicated by. In this paper, all weights are effectively binary, thogh is represented with 27 and with. Hence, the maximm vale possible in any cell when I7 is presented is 2x27=524. The model in this example also assmes that all inpts will have exactly 2 active pixels (this can be relaxed bt is assmed for simplicity). is the normalized vale, as in Eq. 2, where π = 2. We assme λ = here, ths by Eq. 3, V i= i. A cell s V vale represents the total local evidence that it shold be activated. However, rather than simply picking the max V cell in each CM as winner (i.e., hard max), which wold amont to execting only steps -4 of the CSA, the remaining CSA steps, 5-, are exected, in which the V distribtions are transformed as described earlier and winners are chosen via soft max in each CM [final winner choices, chosen from the ρ distribtions are shown in the row of triangles jst below CM indexes]. Ths, an extremely cheap-to-compte (CSA Step 5) global fnction of the whole mac, G, is sed to inflence the local decision process in each CM. We repeat for emphasis that no part of the CSA explicitly operates on, i.e., iterates over, stored hypotheses; indeed, there are no explicit (localist) representations of stored hypotheses on which to operate. Fig. 4 shows that different inpts yield different likelihood distribtions that correlate approximately with similarity. Inpt I 8 (Fig. 4a) has highest intersection with I 2 and a different pattern of intersections with the other learned inpts as well (refer to Fig. 3a). Fig. 4c shows that the codes of the stored inpts become active in approximate proportion to their similarities with I 8, i.e., their likelihoods are simltaneosly physically represented by the fractions of their codes which are active. The G vale in this case,.65, yields, via CSA steps 6-8, the V-to- transform shown in Fig. 4b, which is applied in all CMs. Its range is [,3] and given the particlar V distribtions shown in Fig. 4d, the cell with the max V in each CM ends p being greatly favored over other lower-v cells. The red box shows the V distribtion for CM 9. The second row of the abscissa in Fig. 4b gives the within-cm indexes of the cells having the corresponding (red) vales immediately above (shown for only three cells). Ths, cell 3 has V=.74 which maps to approximately 25 whereas its closest competitors, cells 4 and 6 (gray bars in red box) have V=.9 which maps to =. Similar statistical conditions exist in most of the other CMs. However, in three of them, CMs,, and 4, there are two cells tied for max V. In two, CMs and 4, the cell that is not contained in I 2 s code, φ(i 2), wins (red triangle and bars), and in CM, the cell that is in φ(i 2) does win (black triangle and bars). Overall, presentation of I 8 activates a code φ(i 8) that has 2 ot of 24 cells in common with φ(i 2) manifesting the high likelihood estimate for I 2. To finish or demonstration for the spatial inpt case, Fig. 4e shows presentation of another inpt, I 9, having half its pixels in common with I 3 and the other half with I 6. Fig. 4g shows that the codes for I 3 and I 6 have both become approximately eqally (with some statistical variance) active and are both more active than any of the other codes. Ths, the model is representing that these two hypotheses are the most likely and approximately eqally likely. The exact bar heights flctate somewhat across trials, e.g., sometimes I3 has higher likelihood than I 6, bt the general shape of the distribtion is preserved. The remaining hypotheses likelihoods also approximately correlate with their pixelwise intersections with I 9. The qalitative difference between presenting I 8 and I 9 is readily seen by comparing the V rows of Fig. 4d and 4h and seeing that for the latter, a tied max V condition exists in almost all the CMs, reflecting the eqal similarity of I 9 with I 3 and I 6. In approximately half of these CMs the cell that wins intersects with φ(i 3) and in the other half, the winner intersects with φ(i 6). In Fig. 4h, the three CMs in which there is a single black bar, CMs, 7, and 2, indicates that the codes, φ(i 3) and φ(i 6), intersect in these three CMs. 2

13 φ(i 8 ) CM 9 Likelihood (L) I 8 I I 2 I 3 I 4 I 5 I 6 a) b) V c) d) CM ρ V= φ(i 9 ) CM Likelihood (L) I 9 I I 2 I 3 I 4 I 5 I 6 e) f) V g) h) CM ρ V= Fig. 4 Details of presenting two frther novel inpts, I 8 (panels a-d) and I 9 (panels e-h). In both cases, the reslting likelihood distribtions correlate closely with the inpt overlap patterns. Panels b and f show details of one example CM (indicated by red boxes in panels d and h) for each inpt 3

14 Single SDR code represents entire prob. distribtion: Spatiotemporal (seqential) case Or goal in this section is to demonstrate moment-to-moment (frame-by-frame) pdating of the similarity/likelihood distribtion over stored inpts, which in this case are particlar spatiotemporal moments, in approximate agreement with spatiotemporal similarity strctre of the experienced inpts. Fig. 5 (top row) shows the training set consisting of two 4-item seqences, S=[ABCD] and S2=[EFGH], where the items are the 2x2 pixel patterns shown. Fig. 5 (bottom row) shows two novel test seqences, S3 and S4, constrcted from the same or slightly pertrbed versions of the frames comprising the training seqences. We will present the details of testing on S3 and S4, bt begin by showing, as a baseline, the details of testing on a training seqence, S, in Figres 6 and 7. The training seqences were handcrafted to show natralistic edge motion patterns [assming simple preprocessing (edge-filtering, binarization, and skeletonization)] on a 2x2 apertre of the visal field and sch that the degree of pixel overlap amongst the frames is low. The test seqences were constrcted so that their first and second halves wold nambigosly be spatiotemporally most similar, in terms of the raw measre, pixel overlap, to the first and second halves of the training seqences. Ths, the S3 sbseqence [A B] is clearly most similar to S sbseqence [AB], S3 sbseqence [GH] is clearly most similar (in fact identical) to S2 sbseqence [GH], etc. The overall goal of the demonstrations of S3 and S4 is to show that the likelihood distribtion shifts to reflect, approximately, the spatiotemporal similarities of the stored hypotheses as we switch at mid-seqence between slightly noisy/pertrbed versions of the two learned seqences. Train Set S A B C D E F G H S2 From S From S2 From S2 From S Test Set S3 A B G H E F C D Fig. 5 Train and test seqences sed to demonstrate frame-by-frame pdate of the likelihood distribtion for the case of spatiotemporal (seqential) patterns. The prime mark ( ) is sed to indicate that the frame is a noisy/deformed version of the corresponding learning frame Fig. 6 shows the state of the model, i.e., the inpt and the code activated in response, at the for moments of the test presentation of S. A sample of the active afferent (ble) and H (green) weights on each moment (frame) are shown. The cell at the sorce of an H weight will have been active on the prior moment. In panels, c and d, we show, for the selected cell, all weights increased throghot learning: ths, it can be seen that the selected cell was active not jst on the moment depicted [ble lines originating from active (black) pixels] bt on other moments as well [ble lines originating from inactive (white) pixels]. This figre also introdces or moment notation. By moment, we mean a particlar spatial inpt (seqence item) in the context of the fll item seqence (prefix) that preceded it, which we indicate by enclosing the seqence inclding the crrent item is brackets and bolding the crrent item. S4 4

15 a) φ([a]) b) φ([ab]) c) φ([abc]) d) φ([abcd]) [A] [AB] [ABC] [ABCD] Fig. 6 The state of the model at the for moments of the test presentation of S. Fig. 7 shows the processing details when the training seqence S is presented as a test. Note that this model has Q=9 CMs each with K=8 cells. [Note that the and V traces appear separately: while this is redndant for the first item of a seqence, since =V, it is not for all non-initial items, since V=H]. In total, eight spatiotemporal moments were stored in the mac dring learning. S s first moment is the presentation of its first item, A, denoted [A], followed by [AB], [ABC], and [ABCD], as shown in Fig. 6. Similarly, for S2 s moments, [E], [EF], [EFG], [EFGH], which are not shown The main message of Fig. 7 is that, with each sccessive item presented, the likelihoods of the eight stored hypotheses, i.e., the hypothesis that the crrent inpt moment is [A], that it is [AB], etc., are pdated in a way that respects the coarsely ranked spatiotemporal similarity strctre of the experienced inpts. By coarsely ranked, we mean the following. As this is an exact repeat of a training instance, the most likely moment at each time step is the one that occrred dring the training instance. A glance down across the for likelihood charts at right verifies this: the code of the correct moment (ble bar) is activated more strongly than all others (cyan bars). In general, on each time step, the likelihoods of the other seven moments are mch lower, i.e., falling into a second coarse rank. However, the distribtion on the third time step (Fig. 7c) is qite plasibly described as having three ranks, the middle one inclding the bar for moment [E]. This is appropriate and is de to the fact that item E has significant overlap with C (4 pixels, see Fig. 5). When [E] was presented as the first item of S2 s learning trial, the prior learning in the weights that had occrred on the third moment of S s learning trial, [ABC], cased high smmations for the cells in φ([abc]). Since [E] is the first item of S2, no H signals are present and the choice of cells depends only on the inpts, conseqently yielding a relatively high intersection,{ φ([ ABC]) φ([ E]) }. Fig. 7 s charts show that the correct tracking of likelihood is achieved via independent soft max choices in each of the Q CMs. In panels b-d, in all CMs, the correct cell has = and H=, which yields V=. This reflects that fact that the test seqence here is an exact dplicate of one that has been learned, S. Many other cells have either a significant or a significant H vale, reflecting crosstalk de to some cells being involved in the codes of mltiple moments, bt not both. In fact, all incorrect cells appropriately have zero or near-zero V vales. Ths, in all CMs, the ρ distribtion greatly favors the correct cell. Two errors occr, one in CM 7 for moment [A] and one in CM for moment [ABCD]: as winners are picked sing soft max, there are occasional instances in which a far less likely neron is selected. Nevertheless, we see that the effect of global familiarity, G, which eqals one at all for moments, modlates the V-to- transform in sch a way as to case almost the whole stored code to activate correctly, i.e., to increase the correlation of those cells. And, as will be seen in Figres 8 and 9, lower G vales decrease correlation. Ths, or theory provides a novel, casal, indeed normative, explanation of correlation in the brain. The degree of correlation from one moment to the next, both dring learning and retrieval (inference), is effectively modlated (thogh indirectly) by a deterministic mechanism, the modlation of the V-to- transform. Indeed, all CSA steps except for the last, are deterministic. 5

16 a) CM ρ V= b) ρ V=H H h c) ρ V=H H h L L V Likelihood (L) [A] 82 V [A] [AB] [ABC] [ABCD] [E] [EF] [EFG] [EFGH] [ABC] [E] V d) ρ V=H H h Fig. 7 Detailed trace information for test presentation of all for items of S=[ABCD], i.e., a train=test sanity test. Becase S was also a training item and becase only one other seqence has been stored in the mac (and ths, crosstalk is low), the model detects G=. (% familiarity) for each item and reinstates the traces almost perfectly. The V-to- fnction is the same at all for moments becase G= in all for cases. We show the details of the V-to- transform for one of the CMs, CM, at pper right of each panel. The likelihoods over the eight moments stored in the mac are shown at right: the key for the eight moments is the same for all panels and is shown only in panel b to redce cltter. Panel a has no h or H traces since they are not relevant on the first item of a seqence L 82 [ABCD] V 6

17 We now consider two novel seqences as frther evidence that hypothesis likelihoods, measred as active fractions of their SDR codes, track the coarsely-ranked spatiotemporal similarity of the presented seqence from moment to moment. Fig. 8 shows the state of the model at the for moments of both seqences, S3=[A BGH] and S4=[E F C D]. As noted earlier, these seqences were constrcted so that the relative spatiotemporal similarities of the stored (learned) moments and the crrent test moment wold be clear even withot an exact spatiotemporal similarity metric. That is, the first two moments of S3 mst clearly be considered closest to the first two moments of S. The third moment, [A BG] is spatially closest to the third moment of S2, [EFG], bt from a spatiotemporal perspective, i.e., considering the immediately prior two moments as context, the model shold reasonably consider the learned moment, [ABC] to also have elevated likelihood. The likelihood panel in Fig. 9c indeed shows that the two moments, [ABC] and [EFG] have the two highest likelihoods. Their precise likelihoods vary across test instances, bt they are almost always the two highest. This behavior is de to flattened distribtions, reslting from the lower G (=.478) and the fact that in many CMs, the cell that was in φ([abc]) and the cell that was in φ([efg]) are tied or nearly tied for the max V. i. φ([a ]) φ([a B]) φ([a BG]) φ([a BGH]) a) b) c) d) [A ] [A B] [A BG] [A BGH] ii. φ([e ]) φ([e F ]) φ([e F C ]) φ([e F C D]) a) [E ] b) [E F ] c) [E F C ] d) [E F C D] Fig. 8 The state of the model at the for moments of each of the novel test seqences, S3=[A BGH] (i) and S4=[E F C D] (ii) The same reasoning that allows s to consider S3 s third moment, [A BG] ambigos, i.e., taking temporal context into accont, sggests that the forth moment, [A BGH], shold be jdged less ambigos and in fact more likely to be an instance of learned moment, [EFGH]. This is indeed reflected in the likelihood panel in Fig. 9d. The higher global familiarity, G=.566, reslts in a more expansive V-to- transform, which, in combination with higher V vales for the cells in φ([efgh]) reslts in those cells being greatly favored in each CM. Ths, the model is seen to have sccessflly gone throgh an ambigos state of a seqence, recovering via the on-line combining of new evidence to yield an appropriately less ambigos internal state. This example nderscores another crcial capability of the model: namely, that by allowing stored hypotheses to be physically active (in proportion to their code s overlap with the crrently active code), it allows transiently weaker hypotheses to recover based on ftre evidence and contermand transiently stronger hypotheses. For example, in Fig. 9c, the strongest hypothesis, [ABC] is consistent with the overarching hypothesis that the crrently nfolding seqence is [ABCD] despite the inconsistent evidence presented on the third time step, inpt state G, which is more consistent with the overarching hypothesis 7

18 that the nfolding seqence is [EFGH]. However, when additional evidence inconsistent with [ABCD] occrs on the forth time step (Fig. 9d), the overarching hypothesis that the crrently nfolding seqence is [EFGH] becomes strongest. a) b) ρ V= ρ V=H c) H h ρ V=H d) H h ρ V=H H h CM Fig. 9 Detailed trace information for test presentation of all for items of S3=[A BGH] Fig. shows the reslts for the final example, S4=[E F C D] whose first two moments were constrcted to be closest to the first two moments of S2 and whose second two moments are very close to the last two moments of S. The behavior of the model is broadly analogos to its behavior for S3 in that at its third moment, [E F C ], the two learned moments, [ABC] and [EFG], deemed most likely by the model, are, by the same reasoning applied for S3, plasibly spatiotemporally most similar to [E F C ]. L L L L 533 [A] 82 G =.833 V G =. V [A] [AB] [ABC] [ABCD] [E] [EF] [EFG] [EFGH] G =.478 [ABC] V G =.566 V [EFG] [EFGH] 8

19 Global familiarity, G=.34, is lower here than for the third moment of S3 becase we introdced more noise to the first two moments of S4 than to S3. In fact, the spatial inpt E has the same intersection with E as it does with C (as can be seen by close inspection of Fig. 5) and F has 9 of 2 pixels in common with F. Ths, learned moments, [ABC] and [E], have approximately the same likelihood in Fig. a. Althogh the cells comprising φ([abc]) were originally chosen in a spatiotemporal context and therefore had their afferent H weights increased [from the cells comprising φ([ab])], there are no H signals present on the first item of any seqence. Ths, the choice of winners on the first moment of S4 depends only on signals and ths, on the increases made to the weights dring learning. This is why learned moment [ABC] receives a high likelihood at this moment. Fig. b shows that the ambigity present on the first moment is greatly diminished by the presentation of spatial inpt F, which is qite similar to F, ths making the spatiotemporal inpt moment, [E F ], mch more spatiotemporally similar to stored moment, [EF], than to stored moment, [ABCD], or to any other learned moment. On the third moment of S4 (Fig. c), we present spatial inpt C which is mch more similar to C than any other spatial inpt. As was the case for the third moment of S3, this leads to the two learned third moments, [ABC] and [EFG] being approximately eqally likely. However, de to the increased noise present in the first three moments of S4 compared to S3, G is lower here and the likelihoods of these two learned moments, i.e., the fractions of their codes active, are appropriately lower than was the case on the third moment of S3. Finally, we present spatial inpt D on the forth moment, [E F C D]. De to the mltiplication of the signals from D and the H signals from φ([f G C ]), which had appreciable overlap with φ([abc])], the cells comprising φ([abcd]) are highly favored in all CMs and end p winning in most of them, yielding the significantly higher likelihood for [ABCD] than all other moments, as seen in Fig. d. Again, the model is seen to negotiate ambigos moments, pdating its active code from moment to moment, sch that it simltaneosly represents what are plasibly (given its small experience of the world) the most likely hypothesis (or hypotheses) and the fll coarsely-ranked distribtion over hypotheses. It has been pointed ot that this ability, i.e., simltaneosly representing mltiple competing hypotheses, as for example is reqired for representing motion transparency, is problematic for theories which se flly distribted codes, as do the PPC theories (Poget, Dayan et al. 2). It is tre that we do not go into nmerical detail jstifying most of the relative similarities/likelihoods (pair-wise and higher orders) present in the likelihood distribtions of Figres 7, 9, and, nor, for that matter, those present in the spatial examples of Figres 3 and 4. However, or examples do demonstrate coarse correlation of spatial / spatiotemporal similarity with likelihood (measred as active fraction of code). Similar capabilities, sch as being able to store and sccessflly recognize/retrieve large nmbers of complex seqences, in which the same item(s) may occr mltiple times and in varying contexts (e.g., a natral lexicon), have previosly been demonstrated (Rinks 996, Rinks 24). 9

20 a) CM ρ V= b) ρ V=H H h L L G =.565 V [ABC] [E] G =.69 V [A] [AB] [ABC] [ABCD] [E] [EF] [EFG] [EFGH] c) ρ G =.34 V=H H h L G =.455 d) ρ V=H H h Fig. Detailed trace information for test presentation of all for items of S4=[E F C D] L 47 [ABC] 26 V V [ABCD] [EFG] 2