### Abstract

Networks of neurons in the brain encode preferred patterns of neural activity via their synaptic connections. Despite receiving considerable attention, the precise relationship between network connectivity and encoded patterns is still poorly understood. Here we consider this problem for networks of threshold-linear neurons whose computational function is to learn and store a set of binary patterns (e.g., a neural code) as "permitted sets" of the network. We introduce a simple encoding rule that selectively turns "on" synapses between neurons that coappear in one or more patterns. The rule uses synapses that are binary, in the sense of having only two states ("on" or "off"), but also heterogeneous, with weights drawn from an underlying synaptic strength matrix S. Our main results precisely describe the stored patterns that result from the encoding rule, including unintended "spurious" states, and give an explicit characterization of the dependence on S. In particular, we find that binary patterns are successfully stored in these networks when the excitatory connections between neurons are geometrically balanced-i.e., they satisfy a set of geometric constraints. Furthermore, we find that certain types of neural codes are natural in the context of these networks, meaning that the full code can be accurately learned from a highly undersampled set of patterns. Interestingly, many commonly observed neural codes in cortical and hippocampal areas are natural in this sense. As an application, we construct networks that encode hippocampal place field codes nearly exactly, following presentation of only a small fraction of patterns. To obtain our results, we prove new theorems using classical ideas from convex and distance geometry, such as Cayley-Menger determinants, revealing a novel connection between these areas of mathematics and coding properties of neural networks.

Original language | English (US) |
---|---|

Pages (from-to) | 2858-2903 |

Number of pages | 46 |

Journal | Neural computation |

Volume | 25 |

Issue number | 11 |

DOIs | |

State | Published - Nov 15 2013 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Arts and Humanities (miscellaneous)
- Cognitive Neuroscience

### Cite this

*Neural computation*,

*25*(11), 2858-2903. https://doi.org/10.1162/NECO_a_00504

}

*Neural computation*, vol. 25, no. 11, pp. 2858-2903. https://doi.org/10.1162/NECO_a_00504

**Encoding binary neural codes in networks of threshold-linear neurons.** / Curto, Carina; Degeratu, Anda; Itskov, Vladimir.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Encoding binary neural codes in networks of threshold-linear neurons

AU - Curto, Carina

AU - Degeratu, Anda

AU - Itskov, Vladimir

PY - 2013/11/15

Y1 - 2013/11/15

N2 - Networks of neurons in the brain encode preferred patterns of neural activity via their synaptic connections. Despite receiving considerable attention, the precise relationship between network connectivity and encoded patterns is still poorly understood. Here we consider this problem for networks of threshold-linear neurons whose computational function is to learn and store a set of binary patterns (e.g., a neural code) as "permitted sets" of the network. We introduce a simple encoding rule that selectively turns "on" synapses between neurons that coappear in one or more patterns. The rule uses synapses that are binary, in the sense of having only two states ("on" or "off"), but also heterogeneous, with weights drawn from an underlying synaptic strength matrix S. Our main results precisely describe the stored patterns that result from the encoding rule, including unintended "spurious" states, and give an explicit characterization of the dependence on S. In particular, we find that binary patterns are successfully stored in these networks when the excitatory connections between neurons are geometrically balanced-i.e., they satisfy a set of geometric constraints. Furthermore, we find that certain types of neural codes are natural in the context of these networks, meaning that the full code can be accurately learned from a highly undersampled set of patterns. Interestingly, many commonly observed neural codes in cortical and hippocampal areas are natural in this sense. As an application, we construct networks that encode hippocampal place field codes nearly exactly, following presentation of only a small fraction of patterns. To obtain our results, we prove new theorems using classical ideas from convex and distance geometry, such as Cayley-Menger determinants, revealing a novel connection between these areas of mathematics and coding properties of neural networks.

AB - Networks of neurons in the brain encode preferred patterns of neural activity via their synaptic connections. Despite receiving considerable attention, the precise relationship between network connectivity and encoded patterns is still poorly understood. Here we consider this problem for networks of threshold-linear neurons whose computational function is to learn and store a set of binary patterns (e.g., a neural code) as "permitted sets" of the network. We introduce a simple encoding rule that selectively turns "on" synapses between neurons that coappear in one or more patterns. The rule uses synapses that are binary, in the sense of having only two states ("on" or "off"), but also heterogeneous, with weights drawn from an underlying synaptic strength matrix S. Our main results precisely describe the stored patterns that result from the encoding rule, including unintended "spurious" states, and give an explicit characterization of the dependence on S. In particular, we find that binary patterns are successfully stored in these networks when the excitatory connections between neurons are geometrically balanced-i.e., they satisfy a set of geometric constraints. Furthermore, we find that certain types of neural codes are natural in the context of these networks, meaning that the full code can be accurately learned from a highly undersampled set of patterns. Interestingly, many commonly observed neural codes in cortical and hippocampal areas are natural in this sense. As an application, we construct networks that encode hippocampal place field codes nearly exactly, following presentation of only a small fraction of patterns. To obtain our results, we prove new theorems using classical ideas from convex and distance geometry, such as Cayley-Menger determinants, revealing a novel connection between these areas of mathematics and coding properties of neural networks.

UR - http://www.scopus.com/inward/record.url?scp=84887326138&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84887326138&partnerID=8YFLogxK

U2 - 10.1162/NECO_a_00504

DO - 10.1162/NECO_a_00504

M3 - Article

C2 - 23895048

AN - SCOPUS:84887326138

VL - 25

SP - 2858

EP - 2903

JO - Neural Computation

JF - Neural Computation

SN - 0899-7667

IS - 11

ER -