Angular path integration by moving "hill of activity": a spiking neuron model without recurrent excitation of the head-direction system

Abstract

During spatial navigation, the head orientation of an animal is encoded internally by neural persistent activity in the head-direction (HD) system. In computational models, such a bell-shaped "hill of activity" is commonly assumed to be generated by recurrent excitation in a continuous attractor network. Recent experimental evidence, however, indicates that HD signal in rodents originates in a reciprocal loop between the lateral mammillary nucleus (LMN) and the dorsal tegmental nucleus (DTN), which is characterized by a paucity of local excitatory axonal collaterals. Moreover, when the animal turns its head to a new direction, the heading information is updated by a time integration of angular head velocity (AHV) signals; the underlying mechanism remains unresolved. To investigate these issues, we built and investigated an LMN-DTN network model that consists of three populations of noisy and spiking neurons coupled by biophysically realistic synapses. We found that a combination of uniform external excitation and recurrent cross-inhibition can give rise to direction-selective persistent activity. The model reproduces the experimentally observed three types of HD tuning curves differentially modulated by AHV and anticipatory firing activity in LMN HD cells. Time integration is assessed by using constant or sinusoidal angular velocity stimuli, as well as naturalistic AHV inputs (from rodent recordings). Furthermore, the internal representation of head direction is shown to be calibrated or reset by strong external cues. We identify microcircuit properties that determine the ability of our model network to subserve time integration function.

Figure 1.

Network architecture. A, The network consists of three cell populations: one excitatory (square; E) and two inhibitory populations (circle; I₁, rightward-projecting population; I₂, leftward-projecting population). Cells are labeled by their preferred directions. The two inhibitory populations form mirror copies of each other. No recurrent excitation between cells in the E population. Excitatory cells send strongest excitation to neurons with the same preferred direction in I₁ and I₂ populations. Inhibitory cells in the I₁ population send strongest inhibition to excitatory cells on the right side, whereas neurons in the I₂ population send strongest inhibition to excitatory cells on the left side. Connections between inhibitory neurons are not shown. All cells in the E population receive external excitatory inputs (data not shown) simulated as un-correlated Poisson spike trains with fixed rates. Afferent inputs carrying AHV information are modeled as uncorrelated Poisson spike trains to cells in the I₁ and I₂ populations. The rates of the Poisson spike trains are the same for neurons in the same population (b₀ + b₁ for the I₁ population; b₀-b₁ for the I₂ population). The AHV-modulated component b₁ is proportional to the rat's AHV, whereas the AHV-independent component b₀ is the rate when AHV = 0. B, Weight functions of the connections shown in the top. Solid line, I₁→E; dashed line, I₂→E; dotted line, E→I₁ (E→I₂).

Figure 2.

Self-sustained persistent activity without recurrent excitation. A stationary hill of activity emerges when the AHV-modulated component of the afferent input is zero (b₁ = 0). The peak of the hill of activity is equally likely to appear at any direction (the peak is ∼180° for the example shown here). A, Traces of the membrane potential of three cells from the excitatory population, with preferred directions 134°, 180°, and 227°, respectively. B, Left, Rastergram of the excitatory population (top) and two inhibitory populations (middle and bottom). Abscissa, Time; ordinate, neurons labeled by their preferred directions. Each dot represents a spike. The thick black line in the top plot is the center of the activity profile calculated by the population vector method. Right, Bell-shaped network activity profiles average over a time period of 6 s. C, Synaptic current distribution across three cell populations. From left to right, Synaptic current profiles for the E, I₁, and I₂ populations, respectively. Red, External current; magenta, recurrent NMDA current; green and blue, recurrent inhibitory currents from the I₁ and I₂ populations; black, total current. All currents, including external, recurrent excitatory, and total inhibitory currents (sum of the inhibitory currents from the I₁ and I₂ populations), are symmetric with respect to the center of the hill of activity, which is ∼180°. I_n (C) - I_syn is plotted.

Figure 3.

Moving hill of activity. The spatially tuned firing pattern propagates at a constant speed when afferent input is non-zero but constant (b₁ = -200 Hz). A, Membrane potential trace of three cells from the excitatory population, with preferred directions 90°, 180°, and 270° respectively. B, Rastergram of the excitatory population (top) and two inhibitory populations (middle and bottom). Abscissa, Time; ordinate, neurons labeled by preferred direction. Each dot represents a spike. The thick black line in the top plot is the peak of the activity profile calculated by population vector method, which is used as the internal representation of the head direction. The peak of the hill of activity is moving with a constant speed of ∼489°/s. C, Network synaptic current profiles of the moving hill of activity. From left to right, Profiles for the E, I₁, and I₂ populations. Red, External current; magenta, recurrent NMDA current; green and blue, recurrent inhibitory current from the I₁ and I₂ populations; black, total current. All of the synaptic currents are calculated at time t = 750 ms (dotted line in top plot of B), when the peaks of the hill of activity are ∼180°. For excitatory cells, the synaptic current profiles from the two inhibitory populations (green vs blue) are asymmetric with respect to the peak of the hill of activity, as is the total current. I_n (C) - I_syn is plotted.

Figure 4.

Traveling velocity of the hill of activity is proportional to the afferent input b₁, in a wide range of moving speeds. Ordinate, Moving speed of the activity hill; abscissa, AHV-modulated afferent input b₁. The speed of the activity hill is measured by the speed of the peak of the excitatory activity hill, which is estimated by a population vector method. The afferent inputs to the two inhibitory populations (rates of the external Poisson spike trains) are b₀ + b₁ and b₀ - b₁, respectively (see also Fig. 1). Here, b₀ is the AHV-independent component (set to 1800 Hz), and b₁ is the AHV-modulated component. When b₁ = 0, the hill of activity is stationary (speed is 0; see Fig. 2). When|b₁| ≤ 0.4, the traveling velocity depends on b₁ approximately linearly (with a slope of approximately -2511° · s^-1 · kHz^-1). When|b₁| ≥ 0.7, the speed saturates (with a maximal speed of ∼1670°/s).

Figure 10.

Integration of the angular head velocity profile recorded from a freely moving rat (data provided by J. Taube). A, Experimental AHV profile to be integrated by the model network (left) and its histogram (right). B, The model network integrates the naturalistic AHV profile fairly well. Thick gray line, Experimental HD profile; black lines, traces of the excitatory activity hill in 20 trials. Abscissa, Time; ordinate, head direction. C, Left, Time evolution of the integration error in 20 trials (black lines) and their mean (gray line). Right, The variance of the integration error across 20 simulations increases with time.

Figure 5.

Angular head velocity modulation of the head-direction tuning curves. From left to right, neurons are chosen from the E, I₁, and I₂ populations, respectively. A, Head-direction tuning curves under different angular head velocity. Each curve was obtained from simulation with fixed afferent input b₁. B, The peak firing rate as a function of angular head velocity. The cell from the excitatory population shows symmetric AHV modulation, whereas the two cells from the two inhibitory populations show asymmetric angular head velocity modulation. Cells from the two inhibitory populations behave in the opposite way: one prefers clockwise turning, whereas the other prefers counterclockwise turning. Note that the velocity sensitivity of the inhibitory neurons (middle and right columns) is approximately three times of that of the excitatory neurons (left column). All three types of AHV modulation of HD tuning curves have been observed experimentally. C, The width of the tuning curves as a function of angular head velocity. For the excitatory cell (left), the width increases when the amplitude of the AHV increases. For inhibitory cells, the width increases when AHV increases on one direction but decreases on the other. Neurons from different inhibitory populations (middle vs right columns) behave in the opposite way. Note that the velocity sensitivity of the inhibitory neurons (middle and right columns) is approximately twice of that of the excitatory neurons (left column).

Figure 6.

Cross-inhibition is a key requirement for generating stationary hill of activity. A, Network activity profiles as a function of θ_I (offset angle of the inhibitory-to-excitatory connections, ), from left to right, and σ_E (width of the excitatory-to-inhibitory connections, ), from from top to bottom. B, Height of the network activity profile increases with θ_I. Different labels correspond to different values of σ_E. θ_I = 0 corresponds to iso-inhibition (excitatory neurons near the peak of the activity hill receive strong inhibitory feedback from nearby inhibitory neurons), whereas large θ_I means cross-inhibition (inhibitory cells at the peak of the activity hill inhibit excitatory cells on the flanks of the activity hill). When θ_I is decreased, inhibition change gradually from cross-directional to iso-directional, which results in the decrease of the height of the hill of activity. Hill of persistent activity is possible only for θ_I above a certain critical value (below which the iso-inhibition demolishes hill of activity).

Figure 7.

Angular path integration depends on the offset parameter θ_I of cross-inhibition. A, Comparison of three parameter sets (indicated by a-c in Fig. 6). For each parameter set, the AHV-b₁ relationship was computed, of which the slope of the linear portion of the curve (bottom) and the saturation speed (top) were extracted. When comparing two cases (a and b) with the same offset θ_I but different σ_E (black vs gray), both the slope and the saturation speed are similar. On the other hand, comparing two cases (a and c) with similar stationary hill of activity but realized by two different θ_I values (gray and white), the saturation speed is much lower with a larger θ_I. B, Both the saturation speed (v_sat) and the slope of the AHV-b₁ curve decrease with θ_I linearly. All the other parameters are the same (σ_E = 135°).

Figure 8.

Dependence on the NMDA/AMPA ratio at the excitatory-to-inhibitory connections. For both stationary activity hills (A) and moving activity hills (B), random drifts decrease with the NMDA/AMPA ratio (100% left column; 50% right column). Population vectors calculated from 20 trials are shown. C, The AHV-b₁ relationship for the network with a 50% NMDA/AMPA ratio. The saturation speed is ∼2760°/s, and the slope of the AHV-b₁ curve (linear regression of data points with|b₁| ≤ 0.35) is approximately -3791° · s^-1 · kHz^-1. Both the saturation speed and the slope is larger than for the network with 100% NMDA. D, Drifts are comparable for activity hills moving at different speeds, in both the network with 100% NMDA (ordinate) and the network with a 50% NMDA/AMPA ratio (abscissa). Open symbols, Drift variance at t = 3 s; filled symbols, drift variance at t = 6s. Variance was calculated from population vectors of at least 100 trials.

Figure 9.

Anticipatory time can be accounted for with the inclusion of an acceleration-modulated component in the afferent input b₁. A, Top, Sinusoidal angular head velocity profile to be integrated by the network. The period is 2 s, and the maximal AHV is 300°/s. Bottom, The corresponding angular acceleration profile. B, With an AHV-modulated afferent input, the network is able to integrate the sinusoidal AHV profile accurately with the introduction of a time constant τ_b for the afferent input b₁ (bottom). Top,τ_b = 0; bottom,τ_b = 25 ms. Solid line, The peak of the excitatory activity hill; dashed line, the mathematical integral of the sinusoidal function in A. C, Top, By introducing an acceleration-modulated component to the afferent input b₁ (with coefficient τ₁ = 50 ms), the network (solid line) anticipates future HD (dashed line) by 50 ms. Bottom, The ATI increases linearly with the coefficient of the acceleration-modulated component τ₁. For the result shown in this panel, we used τ_b = 25 ms. ACC, Angular acceleration.

Figure 11.

Calibration by landmark. Landmark stimulus is simulated as a Gaussian-shaped current injection centered at the arrow (from top to bottom, 90°, 145°, and 180° away from the peak of the original hill of activity), and each column corresponds to a different value of stimulus strength (from left to right, 0.1, 0.3, and 0.5 nA). Abscissa, Time; ordinate, excitatory cells labeled by their preferred directions. Each dot represents one spike. The landmark stimulus is applied between the dashed lines (duration of 0.5 s). Depending on the strength and location of the stimulus, the hill of activity behaves in one of two modes: time integration (top left) or winner-take-all (bottom right).