Hippocampal Spike-Timing Correlations Lead to Hexagonal Grid Fields

Abstract

Space is represented in the mammalian brain by the activity of hippocampal place cells as well as in their spike-timing correlations. Here we propose a theory how this temporal code is transformed to spatial firing rate patterns via spike-timing-dependent synaptic plasticity. The resulting dynamics of synaptic weights resembles well-known pattern formation models in which a lateral inhibition mechanism gives rise to a Turing instability. We identify parameter regimes in which hexagonal firing patterns develop as they have been found in medial entorhinal cortex. PACS numbers: 87.19.lv,87.10.Ed,02.30.Jr The spatial position of an animal can be reliably decoded from the neuronal activity of several cell populations in the hippocampal formation [1][2][3]. For example, place cells in the hippocampus fire at only few locations in a spatial environment [4,5] and the position of the animal can be readily read out from single active neurons. Grid cells of the medial entorhinal cortex (MEC) fire at multiple distinct places that are arranged on a hexagonal lattice [6,7]. Although hexagonal patterns are abundant in nature and there exist well-studied physical theories for their emergence, the mechanistic origin of this neuronal grid pattern is still unclear. Initially it was suggested that they result from continuous attractor dynamics [8,9] or superposition of plane wave inputs [10] and, based on circuit anatomy, place cells would then result from a superposition of many grid cells [11,12]. More recent experiments, however, reported place cell activity without intact grid cells, such that grid cells are not the unique determinants of place field firing [13][14][15][16][17]. Conversely, it would thus be possible that grid fields may arise from place field input as suggested in [18][19][20]. The biological mechanisms proposed by these latter theories, however, remain hypothetical. In the present Letter, we propose a learning rule for grid cells based on the individual spike timings of place cells using spike-timing dependent synaptic plasticity (STDP) [21][22][23]. The theory thereby predicts that the observed temporal hippocampal firing patterns (phase precession and thetascale correlations; see below) [24][25][26] translate the temporal proximity of sequential place field spikes into spatial neighborhood-relations observed in grid-field activity. For our model to work, we only have to assume that the synaptic plasticity rule averages over a sufficiently long time interval.Model. We use the classical formulation of pairwise additive STDP [22,27], where the update of a synaptic weight J n , n = 1, . . . , N at time t is computed as [22] C n (s) denotes the time averaged correlation function between the spike train of presynaptic neuron n and the postsynaptic neuron, the learning window W (s) describes the update of the synaptic weight as a function of the time difference s between a pair of pre-and postsynaptic action potentials, and the function F implements soft bounds for the weight increase. The dynamics is further co...

Recurrent amplification of grid‐cell activity

“…A central assumption of our work is that grid‐cell activity originates via a feedforward mechanism prior to the development of the recurrent connections (e.g., Castro & Aguiar, 2014; D'Albis & Kempter, 2017; Dordek et al, 2016; Mhatre et al, 2012; Monsalve‐Mercado & Leibold, 2017; Stepanyuk, 2015; Weber & Sprekeler, 2018). Here, we do not model such a feedforward mechanism explicitly, but we posit that feedforward connections from spatially selective inputs (e.g., place cells or spatially irregular cells) have been already tuned to produce noisy grid patterns.…”

“…Alternatively, grid‐cell activity could arise in a feedforward network prior to the development of the recurrent connections (e.g., Castro & Aguiar, 2014; D'Albis & Kempter, 2017; Dordek, Soudry, Meir, & Derdikman, 2016; Mhatre, Gorchetchnikov, & Grossberg, 2012; Monsalve‐Mercado & Leibold, 2017; Stepanyuk, 2015; Weber & Sprekeler, 2018). In this case, single‐cell grids may spontaneously emerge via three simple ingredients: (a) spatially tuned feedforward inputs; (b) Hebbian synaptic plasticity; and (c) a cell‐intrinsic mechanism that generates spatial periodicity, for example, firing‐rate adaptation (D'Albis & Kempter, 2017; Kropff & Treves, 2008), phase precession (Monsalve‐Mercado & Leibold, 2017), or excitation/inhibition balance (Weber & Sprekeler, 2018). A strength of these models is that they can explain how a grid‐cell circuit develops in a self‐organized manner.…”

A Brain-Inspired Adaptive Space Representation Model Based on Grid Cells and Place Cells

“…where d is the minimum distance from the exploring location to the boundary; l min is the minimum standard deviation of the firing field size; l max is the maximum standard deviation of the firing field size; L is the firing field distribution constant; D is the maximum distance from which visual place cells can be generated. e feedforward network based on place cell inputs and Hebbian learning to weights can be used to generate grid cell with hexagonal firing field [41][42][43][44][45][46]. e periodic grid cell firing field is derived from the periodic weight distribution from place cells to single grid cell, and the input correlation driving the development of periodic weight distribution is usually presented as the Mexican hat model.…”

A Generalized Linear Model of a Navigation Network

“…While most experimental efforts devoted to the neural circuit underlying navigation have explored single cell properties, efforts to understand the mechanism of these cellular properties tend to remain on the network level of description. This has resulted in the development of several theoretical models (Zhang, 1996;Fuhs, 2006;McNaughton et al, 2006;Kropff and Treves, 2008;Burak and Fiete, 2009;Couey et al, 2013;Stepanyuk, 2015;Dordek et al, 2016;D'Albis and Kempter, 2017;Monsalve-Mercado and Leibold, 2017;Weber and Sprekeler, 2018) designed to account for the cellular properties of grid and head direction cells in terms of the architecture of neural networks.…”