Noise promotes independent control of gamma oscillations and grid firing within recurrent attractor networks
- University of Edinburgh, United Kingdom
- Institute for Adaptive and Neural Computation, United Kingdom
Figures
Figure 1 with 1 supplement
Attractor network model with feedback inhibition and theta frequency inputs.
(A) A schematic of populations of excitatory cells (E cells, red) and inhibitory cells (I cells, blue) on a twisted torus of size 34 × 30 neurons. The synaptic coupling between the two populations …
Figure 1—figure supplement 1
Synaptic weights in scaled and probabilistic variants of the network.
(A) Output (top) and input (bottom) synaptic weights of an E (left) and I (right) neuron in the middle of the twisted torus in a network in which synaptic weights are scaled according to the …
Figure 2 with 4 supplements
Noise increases the range of synaptic strengths that support grid firing.
(A–C) Example spatial firing fields (left) and spatial autocorrelation plots (right) of E and I cells for networks without noise (A; σ = 0 pA), with noise level set to σ = 150 pA (B), and noise …
Figure 2—figure supplement 1
Sensitivity of grid firing to changes in feedback inhibition, excitation and noise levels in networks with connection probability between pairs of neurons drawn according to the synaptic profile functions in Figure 1B.
(A–C) Example spatial firing fields (left) and spatial autocorrelation plots (right) of E and I cells for networks without noise (A; σ = 0 pA), with noise set to σ = 150 pA (B), and noise set to σ = …
Figure 2—figure supplement 2
Spatial information and sparsity of firing fields of E and I cells.
(A) Spatial information of E (top) and I (bottom) cells as a function of gE and gI in networks from Figure 2. (B) Same as (A), but the color plots show spatial sparsity of E and I cells. Black lines …
Figure 2—figure supplement 3
Gridness scores of I cells.
Colour plots show gridness score as a function of gE and gI for networks without noise (A), with noise standard deviation σ = 150 pA (B), and σ = 300 pA (C). Data are from simulations of networks …
Figure 2—figure supplement 4
Spatial firing fields in networks with uncorrelated spatial input applied to each I cell.
(A) Examples of firing fields of E and I cells. Gridness score and maximal firing rate of the firing field is indicated in the top left and right parts of each firing field, respectively. (B) …
Figure 3 with 4 supplements
Differential sensitivity of gamma oscillations and grid fields to changes in the strength of E and I synapses.
(A–C) Examples of inhibitory (red) and excitatory (blue) synaptic currents recorded respectively from excitatory and inhibitory neurons from simulations highlighted by arrows in panels (D–F). (D–F) …
Figure 3—figure supplement 1
Sensitivity of gamma oscillations to changes in the strength of E and I synapses in networks with connection probability between pairs of neurons drawn according to the synaptic profile functions in Figure 1B.
(A–C) Examples of inhibitory (red) and excitatory (blue) synaptic currents recorded respectively from excitatory and inhibitory neurons from simulations highlighted by arrows in panels (D–F). (D–F) …
Figure 3—figure supplement 2
Scatter plots of gridness score as a function of the amplitude of gamma oscillations.
(A–C) The plots show relationships between grid field computations (gridness score) and the power of nested gamma oscillations for deterministic networks (A), networks with moderate noise (B) and …
Figure 3—figure supplement 3
Scatter plots of gridness score as a function of the detected oscillation frequency.
(A–C) The plots show relationships between grid field computations (gridness score) and the frequency of gamma oscillations for deterministic networks (A), networks with moderate noise (B) and …
Figure 3—figure supplement 4
Amplitude and frequency of gamma oscillations in the gE and gI parameter regions where grid fields are robust.
Amplitude (top) and frequency (bottom) of detected gamma oscillations for simulations in which gridness score is greater than 0.5, in deterministic networks (A), networks with an intermediate level …
Figure 4 with 1 supplement
Noise promotes formation of continuous attractors.
(A) Examples of snapshots of network activity of E cells from simulations in which velocity and place cell inputs are inactivated. Each row shows a simulation trial with a value of gE and gI …
Figure 4—figure supplement 1
Sensitivity of bump attractor spontaneous drift to variations in gE and gI and noise levels.
(A) Schematic of the bump attractor drift estimation procedure. The first 500 ms of a simulation trial are used to initialize the bump attractor. Onset of theta modulated input current was at 500 …
Figure 5 with 1 supplement
Noise opposes generation of seizure-like states.
(A–C) Raster plots show activity of all neurons in the excitatory (red) and inhibitory (blue) populations for the duration of two theta cycles (top), along with the average population firing rates …
Figure 5—figure supplement 1
Examples of activity in the network.
(A–C) Top: Mean maximal firing rate per theta cycle (average over five trials), outlining the average activity during theta cycles, in the parameter space of gE and gI. Center and bottom: Raster …
Figure 6 with 5 supplements
Seizure-like states and grid firing fields in networks without theta frequency inputs.
(A–C) Maximal average population firing rate of E cells estimated from the whole simulation run (10 s; 500 ms at the beginning of the simulation excluded) for each simulated level of noise indicated …
Figure 6—figure supplement 1
Effect of replacing theta frequency inputs by a constant input with an equal mean amplitude.
(A–C) Amplitude (top) and frequency (bottom) of detected gamma oscillations (‘Materials and methods’) in deterministic networks (A), networks with an intermediate level of noise (B) and in networks …
Figure 6—figure supplement 2
Effect of noise on gridness scores in networks without theta frequency inputs.
The plot shows a difference between gridness scores of networks with σ = 150 pA and networks with σ = 0 pA plotted as a function of gE and gI when theta inputs were replaced with a constant input …
Figure 6—figure supplement 3
Firing rates of E cells.
(A) Average firing rate of all E cells during simulations of animal movement as a function of gE and gI. Black lines outline the region from Figure 2D–F where gridness score = 0.5. (B) Relationship …
Figure 6—figure supplement 4
Calibration of the gain of the velocity inputs.
(A–C) Bump attractor speed as a function of the strength of the velocity current for the three simulated levels of noise. 10 simulation runs were performed for each level of noise (blue markers). In …
Figure 6—figure supplement 5
Effectivity of the place cell resetting mechanism as a function of gE and gI and noise levels.
(A) Illustration of the procedure to estimate the difference between the bump position induced by place cells and actual estimated position of the bump state, by using a sliding window with 250 ms …
Figure 7 with 10 supplements
Gridness scores and gamma activity in networks with recurrent inhibition.
(A–C) Gridness score as a function of gE and gI for networks without noise (A; σ = 0 pA), with noise level set to σ = 150 pA (B), and noise level set to σ = 300 pA (C). Simulations that did not …
Figure 7—figure supplement 1
Spatial firing fields in networks that contain recurrent I → I synapses.
(A–C) Example spatial firing fields (left) and spatial autocorrelation plots (right) for networks with gE = 3 nS and gI = 1 nS (A) and networks with gE = 1 nS and gI = 3 nS (B), corresponding to the …
Figure 7—figure supplement 2
Continuous attractors in networks that contain direct I → I synapses.
(A) Examples of E cell population firing rate snapshots from simulations in which velocity and place cell inputs are inactivated. Each row shows a simulation trial with a value of gE and gI …
Figure 7—figure supplement 3
Sensitivity of bump attractor spontaneous drift to variations in gE, gI and noise levels in networks that contain direct I → I synapses.
(A) Schematic of the bump attractor drift estimation procedure. The first 500 ms of a simulation trial are used to initialize the bump attractor. Onset of theta modulated input current was at 500 …
Figure 7—figure supplement 4
Calibration of the gain of the velocity inputs in networks that contain direct I → I synapses.
(A–C) Bump attractor speed as a function of the strength of the velocity current for the three simulated levels of noise indicated by σ. Values of gE and gI are indicated by arrows in (D–I). 10 …
Figure 7—figure supplement 5
Seizure-like states in networks that contain direct I → I synapses.
(A–C) Raster plots show activity of all neurons in the excitatory (red) and inhibitory (blue) populations for the duration of two theta cycles (top), along with the average population firing rates …
Figure 7—figure supplement 6
Sensitivity of grid firing to changes in inhibition and excitation in networks that contain direct E → E synapses.
(A–C) Example firing fields (left) and spatial autocorrelation plots (right) for the strengths of recurrent synaptic connections indicated by arrows in (D–F) for networks without noise (A; σ = 0 …
Figure 7—figure supplement 7
Sensitivity of gamma oscillations to changes in inhibition and excitation in networks that contain direct E → E synapses.
(A–C) Examples of inhibitory (red) and excitatory (blue) synaptic currents recorded respectively from excitatory and inhibitory neurons from simulations highlighted by arrows in panels (D–F). (D–F) …
Figure 7—figure supplement 8
Continuous attractors in networks that contain direct E → E synapses.
(A) Examples of E cell population firing rate snapshots from simulations in which velocity and place cell inputs are inactivated. Each row shows a simulation trial with a value of gE and gI …
Figure 7—figure supplement 9
Seizure-like states in networks that contain direct E → E synapses.
(A–C) Raster plots show activity of all neurons in the excitatory (red) and inhibitory (blue) populations for the duration of two theta cycles (top), along with the average population firing rates …
Figure 7—figure supplement 10
Probability of bump formation and network activity plots in networks with structured E → E and unstructured E → I and I → E connections.
Since the presence of bump attractors is necessary for grid computation, we tested whether networks with only structured E-E connections can generate activity bumps. We used the Gaussian fitting …
Appendix figure 1
(A) Histogram of velocities of a simulated animal.
(B) Histogram of bump speeds derived from the animal velocities estimated in Equation 21, for different grid field spacings.
Tables
Appendix table 1
Parameter values for synapses
| Name | Units | Value |
|---|---|---|
| EAMPA | mV | 0 |
| τAMPA | ms | 1 |
| ENMDA | mV | 0 |
| τNMDA | ms | 100 |
| mV | −75 | |
| ms | 5 |
Appendix table 2
Neuron parameters and their description
| Name | Description | Name | Description |
|---|---|---|---|
| Vm | Membrane potential | EAMPA | AMPA reversal potential |
| Cm | Membrane capacitance | gNMDA | NMDA conductance |
| gL | Leak conductance | ENMDA | NMDA reversal potential |
| EL | Leak reversal potential | Im | Trans-membrane current |
| gAHP | AHP conductance | Isyn | Synaptic current |
| τAHP | AHP time constant | Isyn | Synaptic current |
| EAHP | AHP reversal potential | Iext | External current |
| ΔT | Spike initiation width | Iconst | Constant current |
| VT | Spike initiation threshold | Iθ | Theta-modulated current |
| GABA conductance | Ivel | Velocity current | |
| GABA reversal potential | Iplace | Place cell current | |
| gAMPA | AMPA conductance | τAMPA | AMPA time constant |
| GABA time constant | τNMDA | NMDA time constant | |
| gad | Adaptation conductance | τad | Adaptation time constant |
| AHP maximal value | Adaptation conductance increase | ||
| Aθ | θ-current amplitude | fθ | θ-current frequency |
| ϕθ | θ-current phase | – | – |
| wAMPA | AMPA synaptic weight | wNMDA | NMDA synaptic weight |
| GABA synaptic weight | – | – |
-
For the exact values used in the simulations, refer to Appendix tables 1, 3–5.
Appendix table 3
Single neuron parameter values for all cells
| Name | Units | Value (E cells) | Value (I cells) |
|---|---|---|---|
| Cm | pF | 211.389 | 227.3 |
| EL | mV | −68.5 | −60 |
| VT | mV | −50 | −45 |
| Vr | mV | −68.5 | −60 |
| gL | nS | 22.73 | 22.73 |
| ΔT | mV | 0.4 | 0.4 |
| EAHP | mV | −80 | × |
| τAHP | ms | 20 | × |
| nS | 5 | × | |
| τad | ms | × | 7.5 |
| nS | × | 22.73 |
Appendix table 4
Parameter values for external inputs
| Name | Units | Value (E cells) | Value (I cells) |
|---|---|---|---|
| Iconst | pA | 300 | 200 |
| Aθ | pA | 375 | 25 |
| ϕθ | rad | −π/2 | −π/2 |
| fθ | Hz | 8 | 8 |
Appendix table 5
Parameter values for synaptic profiles
| Name | Units | Value |
|---|---|---|
| μ | normalised | 0.433 |
| σexc | normalised | 0.0834 |
| σinh | normalised | 0.0834 |
| C | normalised | 0.03 |
| λgrid | cm | 60 |
Additional files
-
Supplementary file 1
Examples of spatial firing fields. (A-L) Top: Gridness score in the parameter space of the E and I synaptic strength scaling parameters (gE and gI respectively). Bottom: Firing fields of a single cell obtained by simulating animal movement, in the parameter region highlighted by black rectangle in the parameter space plot. Above each firing field is the estimated gridness score (left) and maximal firing rate in the firing field (right). Blank (white) locations in the parameter space are simulations that did not finish in the pre-specified time limit (5 hr). Noise level used in each set of simulations is shown by σnoise. Color scale in the firing field plots ranges from 0 Hz (dark blue) to the maximal firing rate for each of the firing fields (dark red).
- https://doi.org/10.7554/eLife.06444.036
