1D and 2D grid cell network and activity pattern.

(a) Stellate cell intrinsic dynamics. In clockwise direction (1) Post-inihibitory rebound as a result of hyperpolarizing pulse input of increasing intensity. (2) Depolarizing sag in response to increasing hyperpolarizing input. (3) Membrane potential resonance in response to chirp input. inset, normalized impedance for the same trace. (4) Subthreshold oscillations in the membrane potential oscillations. inset, normalized impedance. (b) 2D network structure for four sheets of stellate cells and one sheet of inhibitory interneuron. The opposite edges of each sheet are connected to form a torus. The middle schematic shows the offset in the output from stellate sheet representing the east direction to interneuron sheet. The right schematic depicts center-surround connectivity from interneuron to stellate sheet. The same connectivity is used for mutual connections among interneurons (not shown). (c) 2D network activity of stellate cells moving in east (left) and north (right) direction, along with their autocorrelation (bottom). (d) 1D ring network. Two stellate rings and one interneuron ring with 192 neurons in each ring. The interneuron ring is mutually connected with a centre-surround connectivity (I-I, bottom). The same connectivity is projected onto both the stellate rings (I-S, bottom). The connection from the two stellate rings to the interneuron ring has an offset in opposite directions representing left and right movement (S-I, bottom). (e) Raster plot for 1D network. Top, Stellate ring 1 (R1) with clockwise offset. Middle, Interneuron ring. Bottom, Stellate ring 2 (R2) with counterclockwise offset. For the first 2 seconds, both stellate rings receive DC inputs. Then, Stellate R1 receives DC input from 2-4 seconds, followed by Stellate ‘R2 from 4-6 seconds. (f) Grid scale and field sizes can be increased (bottom) by increasing the distance between two peaks of the center-surround connectivity (top)

Path Integration in 1D network.

(a) Left, speed of the network in response to increasing DC inputs to the stellate ring (in blue) and Velocity integration (V-I) curve used for path integration (in orange). Notice that the V-I curve is the inverse of the attractor speed-Input DC relation. Right, the resulting relation between attractor speed and input speed after applying the Velocity integration curve. (b) Path integration of a randomly generated circular trajectory (green). A population vector-based decoder is used to decode positions from the neural activity (blue). Bottom, absolute error for the decoded trajectory at different points in the simulation.

Membrane potential dynamics along grid field.

(a) Membrane potential ramps and theta modulation. Top, Membrane potential ramps (orange) compared with its theta amplitude envelope (green). Spikes demarcating the grid fields are shown in grey. Bottom, Theta amplitude envelope of the overall inhibitory synaptic conductance received by the same cell. (b) Time series histogram of membrane potential theta modulation of all stellate cells. The average is marked in yellow and the location of the grid field is denoted above. (c) Disinhibition for ramp formation. Inhibitory synaptic conductance (in black) to stellate cells shows disinhibition as a mechanism for membrane potential ramp formation (orange). (d) First spike as postinhibitory rebound. Inhibitory synaptic conductance (black) and HCN conductance (blue) along the grid field (gray). Annotations denote the conductance values at the first (green) and the last spike (orange) of the field. At the onset of the field, as the inhibition wanes, the HCN conductance is higher when compared with the end of the field, leading to post-inhibitory rebounds. (e) Phase plot for HCN conductance and membrane potential for 300ms before the onset of the field to the end of the field.

HCN reduces the velocity gain of the ring attractor.

(a) Grid scale vs bump scale. Top, Bump scales and bump sizes are measured along the neurons. The bump scale is a measure of the distance between the centers of two consecutive bumps, and bump size is the number of neurons that constitute a single bump. Bottom, Grid scale and Grid field sizes are measured from recordings of a single neuron along the time (constant speed input) or distance axis. (b) HCN KO shrinks bump sizes. Top, Reduction in bump size in HCN knockout (blue). Bottom, Membrane potential of stellate cells at different locations on the bumps marked in the top panel. Note the drop in resting membrane potential in HCN knockout. At the edges of the bump, the knockout stellate cells are less excitable which leads to a reduction in the bump size. (c) HCN KO lowers inhibition. Average interneuron firing rates as a function of Input DC in HCN wild type (grey), HCN Knockout (blue) and HCN knockout after inhibition is restored (cyan). (d) HCN modulates velocity gain. Attractor speed as a function of input DC. In HCN knockout (blue) higher input DC is required to initiate bump movement, and for most of the input range the same input causes slower bump movement when compared to the wild-type. (e) Grid scale and Grid Field size as a function of Input DC (speed) in HCN wildtype (grey), HCN knockout (blue) and HCN knockout after inhibition is restored (cyan). The range of Input DC used for these simulations is highlighted in d. Bump scales are not affected by HCN knockout. However, Bump sizes shrink due to a drop in the resting membrane potential of stellate cells. Grid scales expand, and grid field sizes expand if the inhibition within the network is restored. In all four cases, the values converge at higher DC inputs.

Predictive coding in the grid cell attractor model.

Neuron intrinsic predictive and retrospective coding – (a) (left) Grid fields from a single stellate cell (averaged across consecutive fields) when the animal is moving either rightward (blue) or leftward (green). The gray curve is the right and left averaged grid field. (right) The same experiment in HCN knockouts abolishes predictive coding. (b) Comparison of inhibitory synaptic conductance received by a stellate cell in the presence (left) and absence (right) of HCN after a single pass through a grid field (in red). The annotations denote the values of the synaptic conductance at the start and the end of the field. (c) HCN density. Positional bias as a result of increasing HCN density. For each HCN density input DC was adjusted to maintain a constant bump velocity (∼30N/s) across all HCN density levels. (d) Onset of first and last spikes. The inhibitory synaptic conductance at the first and last spikes of grid fields is shown at increasing levels of maximal HCN conductance. (e) HCN time scales. The effect of both slow and fast time scales of the HCN conductance on the predictive bias. (f) Speed dependence. Positional bias measured as a function of increasing DC inputs (speed of the animal) in the presence and absence of HCN conductance. Positive values indicate predictive coding and negative values indicate retrospective coding. Network based predictive and retrospective coding – (g) The synaptic profiles of interneuron to stellate connections (clockwise ring) that are used to generate predictive and retrospective bias in the absence of HCN conductance. A mirrored version of the same connection (not shown) is projected to the counterclockwise ring. (h) Top, Network-based predictive bias. As in A, grid fields are shown as the animal is moving in left (green) or right (blue) directions. The gray curve is the right and left averaged grid field. Bottom, Inhibitory synaptic conductance that is received by the same stellate cell. Note the duration of the inhibition is longer at the end of the field. (i) Top, One lobe of the synaptic profile in F (middle) is shown. Asymmetry is generated by adjusting the asymmetry factor κ, which extends the Gaussian on the right side of the peak and shrinks it on the left side. Bottom, The magnitude of the bias as a function of the asymmetry factor in the predictive and the retrospective network. (j) Retrospective bias generated by using the connectivity depicted in the top panel in F.