Combinatorial Neural Codes from a Mathematical Coding Theory Perspective

Curto, Carina; Itskov, Vladimir; Morrison, Katherine; Roth, Zachary; Walker, Judy L.

doi:10.1162/neco_a_00459

Cited by 32 publications

(36 citation statements)

References 52 publications

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“…3f, details vs. time in Supplementary Fig. 3b), suggesting that a combinatorial coding scheme [62][63][64][65] underlies their decoding performance. Together, these findings allude to a neural code that depended little on the precise activity levels of neurons, but more on their identities out of a time-dependent subset of participating neurons.…”

Section: Multiple Present-and Past-trial Task Information Can Be Uniqmentioning

confidence: 99%

Sequential and efficient neural-population coding of complex task information

Koay

Thiberge

Brody

et al. 2019

Preprint

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Recent work has highlighted that many types of variables are represented in each neocortical area. How can these many neural representations be organized together without interference, and coherently maintained/updated through time? We recorded from large neural populations in posterior cortices as mice performed a complex, dynamic task involving multiple interrelated variables. The neural encoding implied that correlated task variables were represented by uncorrelated modes in an information-coding subspace. We show via theory that this can enable optimal decoding directions to be insensitive to neural noise levels. Across posterior cortex, principles of efficient coding thus applied to task-specific information, with neural-population modes as the encoding unit. Remarkably, this encoding function was multiplexed with rapidly changing, sequential neural dynamics, yet reliably followed slow changes in task-variable correlations through time. We can explain this as due to a mathematical property of high-dimensional spaces that the brain might exploit as a temporal scaffold.

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Section: Multiple Present-and Past-trial Task Information Can Be Uniqmentioning

confidence: 99%

Sequential and efficient neural-population coding of complex task information

Koay

Thiberge

Brody

et al. 2019

Preprint

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“…, and Lk 12 (∆| [4] ) is the non-contractible disconnected graph in Figure 4C. Thus, ({1, 2}, {3, 4}) is a local obstruction, and so C cannot be convex.…”

Section: Algebraic Signaturementioning

confidence: 99%

“…Similarly, ({1}, {2, 3, 4}) gives another local obstruction that is not detectable from the canonical form. Specifically, ({1}, {2, 3, 4}) ∈ RF(C) since U 1 ⊆ U 2 ∪ U 3 ∪ U 4 with U 1 ∩ U i = ∅ for each i = 2, 3, 4, and Lk 1 (∆| [4] ) is the non-contractible hollow simplex shown in Figure 4D. In fact, it turns out that every non-maximal σ ∈ ∆ has a related RF relationship that is a local obstruction (see [3, Table 2 in Supplementary Text S1]).…”

Section: Algebraic Signaturementioning

confidence: 99%

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Algebraic signatures of convex and non-convex codes

Curto

Gross

Jeffries

et al. 2019

Journal of Pure and Applied Algebra

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A convex code is a binary code generated by the pattern of intersections of a collection of open convex sets in some Euclidean space. Convex codes are relevant to neuroscience as they arise from the activity of neurons that have convex receptive fields. In this paper, we use algebraic methods to determine if a code is convex. Specifically, we use the neural ideal of a code, which is a generalization of the Stanley-Reisner ideal. Using the neural ideal together with its standard generating set, the canonical form, we provide algebraic signatures of certain families of codes that are non-convex. We connect these signatures to the precise conditions on the arrangement of sets that prevent the codes from being convex. Finally, we also provide algebraic signatures for some families of codes that are convex, including the class of intersection-complete codes. These results allow us to detect convexity and non-convexity in a variety of situations, and point to some interesting open questions.

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“…Interestingly, in exploring the implications of shifting focus from information theory to coding theory in terms of influence upon theoretical neuroscience, (Curto, Itskov et al 2013) have pointed to this same tradeoff, though their treatment uses error rate (coding accuracy) instead of storage capacity. We point out that understanding how neural correlation ultimately affects things like storage capacity is considered largely unknown and an active area of research (Latham 2017).…”

Section: Novel Explanation Of Noise and Correlationmentioning

confidence: 99%

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

Rinkus

2017

Preprint

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The brain is believed to implement probabilistic reasoning and to represent information via population, or distributed, coding. Most previous population-based probabilistic (PPC) theories share several basic properties: 1) continuous-valued neurons (units); 2) fully/densely-distributed codes, i.e., all/most coding units participate in every code; 3) graded synapses; 4) rate coding; 5) units have innate unimodal, e.g., bellshaped, tuning functions (TFs); 6) units are intrinsically noisy; and 7) noise/correlation is generally considered harmful. We present a radically different theory that assumes: 1) binary units; 2) only a small subset of units, i.e., a sparse distributed code (SDC) (a.k.a. cell assembly, ensemble), comprises any individual code; 3) binary synapses; 4) signaling formally requires only single, i.e., first, spikes; 5) units initially have completely flat TFs (all weights zero); 6) units are not inherently noisy; but rather 7) noise is a resource generated/used to cause similar inputs to map to similar codes, controlling a tradeoff between storage capacity and embedding the input space statistics in the pattern of intersections over stored codes, indirectly yielding correlation patterns. The theory, Sparsey, was introduced 20 years ago as a canonical cortical circuit/algorithm model of efficient, generic spatiotemporal pattern learning/recognition, but it was not elaborated as an alternative to PPC-type theories. Here, we provide simulation results showing that the active SDC simultaneously represents not only the spatially/spatiotemporally most similar/likely input but the coarsely-ranked similarity/likelihood distribution over all stored inputs (hypotheses). Crucially, Sparsey's code selection algorithm (CSA), used for both learning and inference, achieves this with a single pass over the weights for each successive item of a sequence, thus performing learning and probabilistic inference for spatiotemporal patterns with a number of steps that remains constant for the life of the system, i.e., as the number of stored items increases. We also discuss our approach as a radically new implementation of graphical probability modeling.

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Combinatorial Neural Codes from a Mathematical Coding Theory Perspective

Cited by 32 publications

References 52 publications

Sequential and efficient neural-population coding of complex task information

Sequential and efficient neural-population coding of complex task information

Algebraic signatures of convex and non-convex codes

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

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