Fig 1.
Continuous attractor networks with any number of bumps can produce head direction cells and grid cells.
(A) Desired tuning curves of a head direction cell and a 1D grid cell. (B) Orientation and position coordinates whose changes drive bump motion. (C) One- and two-bump ring attractor networks. Each black neuron produces the desired tuning curves in A. In the two-bump network, the coupling to coordinate changes is half as strong, and the second bump is labeled for clarity.
Fig 2.
Bump formation in a ring attractor network.
(A) Network schematic with populations L and R and locally inhibitory connectivity W. (B and C) Networks with 200 neurons and 3 bumps. (B) Connectivity weights for a neuron at the origin. The inhibition distance is l = 29 and the connectivity shift is ξ = 2. (C) Steady-state synaptic inputs. Curves for both populations lie over each other. With a ReLU activation function, the firing rates follow the solid portions of the colored lines and are 0 over the dashed portions. The bump distance is λ = 200/3. Thick gray line indicates Eq 4. (D and E) Networks with 500 neurons. (D) More bumps and shorter bump distances are produced by smaller inhibition distances. Points indicate data from 10 replicate simulations. Line indicates Eq 5. (E) The inhibition distance l = 55 corresponds to the black point in D with λ = 125 and M = 4. These values also minimize the Lyapunov functional (Eq 6), which varies smoothly across λ for infinite networks (line) and takes discrete values for finite networks (points). (F) The scaled bump shape remains invariant across network sizes and bump numbers, accomplished by rescaling connectivity strengths according to Eq 7. Curves for different parameters lie over one another.
Fig 3.
Dynamics in a ring attractor network.
(A–C) Networks with 200 neurons and 3 bumps. (A) Synaptic inputs for populations L and R under drive b = 2. Snapshots taken at 150 ms intervals demonstrate rightward motion. (B) Bump velocity is proportional to drive. The connectivity shift is ξ = 2. (C) Bump velocity is largely proportional to connectivity shift. The drive is b = 0.5. (D–H) Networks with synaptic input noise. (D) Bump displacements for 48 replicate simulations demonstrating diffusion with respect to coherent motion. Networks with 200 neurons and 1 bump. (E and F) Mean bump velocity is proportional to drive and remains largely independent of network size, bump number, and noise magnitude. (G and H) Bump diffusion coefficient scales quadratically with noise magnitude, remains largely independent of drive, and varies with network size and bump number. The noise magnitude is σ = 0.5 in D, E, and G, and the drive is b = 0.5 in D, F, and H. Values for both bumps in two-bump networks lie over each other. Points indicate data from 48 replicate simulations and bars indicate bootstrapped standard deviations. Dotted gray lines indicate Eqs 8 and 10.
Fig 4.
Possible mappings between network coordinates and two types of physical coordinates.
(A) In networks encoding linear coordinates such as position, one neuron always represents a fixed physical interval. This mapping is trivial and identical to using network coordinates. (B) In networks encoding circular coordinates such as orientation, the bump distance always represents 360°.
Fig 5.
Bump diffusion due to input and spiking noise.
(A, B) Networks with synaptic input noise of magnitude σ = 0.5 and drive b = 0.5. Dotted gray lines indicate Eq 10. (A) Diffusion decreases with bump number under linear mapping and remains largely constant under circular mapping. Networks with 600 neurons. (B) Diffusion increases with network size under linear mapping and decreases under circular mapping. Networks with 3 bumps. (C and D) Same as A and B, but for networks with Poisson spiking noise instead of input noise. Dotted gray lines indicate Eq 20. Points indicate data from 48 replicate simulations and bars indicate bootstrapped standard deviations.
Fig 6.
Mutual information between neural activity and physical coordinates with input noise of magnitude σ = 0.5.
(A) To compute mutual information, we initialize replicate simulations without input drive at different coordinate values (thick black lines) and record the final neural activities (thin colored lines). The physical coordinate can be linear or circular and its range can be narrow or wide; here, we illustrate two possibilities for networks with 600 neurons and 3 bumps. (B and C) Mutual information between physical coordinate and single-neuron activity under narrow coordinate ranges. (B) Information increases with bump number for linear coordinates and remains largely constant for circular coordinates. Networks with 600 neurons. (C) Information decreases with network size for linear coordinates and increases for circular coordinates. Networks with 3 bumps. (D and E) Mutual information between physical coordinate and single-neuron activity under wide coordinate ranges. The trends in B and C are preserved for circular coordinates. They are also preserved for linear coordinates, except for the shaded regions in which the coordinate range exceeds the bump distance. (F) Coarse local cues are active over different quadrants of the wide coordinate ranges. (G and H) Mutual information between physical coordinate and the joint activities of a single neuron with the four cues in F under wide coordinate ranges. The trends in B and C are preserved for both linear and circular coordinates. Points indicate data from 96 replicate simulations at each coordinate value averaged over neurons and bars indicate bootstrapped standard errors of the mean. Cue icons adapted from Streamline Freemoji (CC BY license, https://www.streamlinehq.com/emojis/freebies-freemojis).
Fig 7.
Bump trapping due to connectivity noise at low drive.
(A–C) Networks with 600 neurons, 1 bump, and the same realization of connectivity noise of magnitude 0.002. (A) Theoretical values for drift velocity as a function of bump position using Eq 24. (B) Bumps drift towards trapped positions over time. The drive is b = 0. Arrows indicate predictions from vconn(θ) crossing 0 with negative slope in A. Lines indicate simulations with different starting positions. (C) Bump trajectories with smallest positive and negative drive required to travel through the entire network. Respectively, b = 0.75 and b = −0.52. The larger of the two in magnitude is the escape drive b0 = 0.75. Note that positions with low bump speed exhibit large velocities in the opposite direction in A. (D and E) Networks with multiple realizations of connectivity noise of magnitude 0.002. (D) Escape drive decreases with bump number under linear mapping and remains largely constant under circular mapping. Networks with 600 neurons. (E) Escape drive increases with network size under linear mapping and remains largely constant under circular mapping. Networks with 3 bumps. Points indicate simulation means over 48 realizations and bars indicate standard deviations. Dotted gray lines indicate Eq 26 averaged over 96 realizations.
Fig 8.
Bump speed irregularity due to connectivity noise at high drive.
(A) Bump speed as a function of bump position with connectivity noise of magnitude 0.002 and drive b = 1.5. Network with 600 neurons, 1 bump, and the same realization of connectivity noise as in Fig 7A–7C. Thick gray lines indicate Eq 25. (B–E) Networks with multiple realizations of connectivity noise of magnitude 0.002 and drive b = 1.5. (B) Speed difference between directions decreases with bump number under linear mapping and remains largely constant under circular mapping. Networks with 600 neurons. (C) Speed difference increases with network size under linear mapping and remains largely constant under circular mapping. Networks with 3 bumps. (D and E) Same as B and C, but for speed variability within each direction. Points indicate simulation means over 48 realizations and bars indicate standard deviations. Dotted gray lines indicate Eqs 30 and 31 averaged over 96 realizations.