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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...

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...

Hippocampal place fields form a neuronal map of the spatial environment. In addition, the distance between two place field centers is proportional to the firing phase difference of two place cells with respect to the local theta rhythm. This consistency between spatial distance and theta phase is generally assumed to result from hippocampal phase precession: The firing phase of a place cell decreases with distance traveled in the place field. The rate of phase precession depends on place field width such that the phase range covered in a traversal of a place field is independent of field width. Width-dependent precession rates, however, generally disrupt the consistency between distance and phase differences. In this paper we provide a mathematical theory suggesting that this consistency can only be secured for different place field widths if phase precession starts at a width-dependent phase offset. These offsets are in accordance with the experimentally observed theta wave traveling from the dorsal to the ventral pole of the hippocampus. Furthermore the theory predicts that sequences of place cells with different widths should be ordered according to the end of the place field. The results also hold for considerably nonlinear phase precession profiles.