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Propagating Activity Patterns in Large-Scale Inhibitory Neuronal Networks

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Science  27 Feb 1998:
Vol. 279, Issue 5355, pp. 1351-1355
DOI: 10.1126/science.279.5355.1351

Abstract

The propagation of activity is studied in a spatially structured network model of γ-aminobutyric acid–containing (GABAergic) neurons exhibiting postinhibitory rebound. In contrast to excitatory-coupled networks, recruitment spreads very slowly because cells fire only after the postsynaptic conductance decays, and with two possible propagation modes. If the connection strength decreases monotonically with distance (on-center), then propagation occurs in a discontinuous manner. If the self- and nearby connections are absent (off-center), propagation can proceed smoothly. Modest changes in the synaptic reversal potential can result in depolarization-mediated waves that are 25 times faster. Functional and developmental roles for these behaviors and implications for thalamic circuitry are suggested.

Reciprocally inhibitory neural circuits are known to be a fundamental substrate for rhythmogenesis in animals' central pattern generator systems (1). Subnetworks of mutually connected neurons were also found in various systems of the mammalian brain, such as the hippocampus, neocortex, and thalamus (2). The operation and significance of such interneuronal networks are not well understood. Recent studies showed that synaptic inhibition can synchronize cells, thereby contributing to the generation of large-scale rhythmic activities observed in the thalamus and hippocampus (3-5). Another possible function is disinhibition, where the interneurons projecting to a principal neuron are selectively inhibited by another group of interneurons. This mechanism would require specialized connectivity patterns between mutually inhibitory cells and with principal neurons (on-center or off-center), which are generally difficult to identify by anatomical and electrophysiological experiments. In the present work, we demonstrate by model simulations that the spatiotemporal activity patterns of an interneuronal population display a number of unique dynamic features. The qualitative characteristics dramatically depend on, and therefore can subserve as predictive indicators of, the underlying synaptic circuit architecture.

As a specific example, the thalamocortical (TC) relay neurons receive powerful recurrent inhibition from GABAergic cells in the thalamic reticular nucleus (RE). How this feedback inhibition is organized spatially will determine in part its role in the sensory information processing within the thalamus (6). The RE-mediated synaptic inhibition is also critically involved in the generation of the synchronous thalamic spindle oscillations during light sleep (3). Moreover, in the thalamic slice models one finds spatiotemporally organized activity in the form of propagating waves, which are waves that move very slowly (∼1 mm/s), on each cycle recruiting additional inactive cells in a distinctive, non-smooth, “lurching” manner (7, 8). In this paper, we consider a spatially structured network of GABAergic neurons, which may be interpreted as a reduction from a two-population thalamic network. The idea is that because the TC-to-RE projection is topographic and acts via rapid glutamate receptors of the α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) type, the excitation in a TC cell would result in a barrage of inhibitory postsynaptic potentials (IPSPs) in the neighboring TC cells (through the disynaptic TC-RE-TC loop). In this idealized view (of the isolated thalamic circuit), the TC cell layer acts effectively as a cell population with reciprocal GABAergic inhibitory interactions.

In our network model, the GABAA-coupled cells have post-inhibitory rebound (PIR) properties and they are not oscillatory (when isolated) for any level of maintained input. A model neuron must be “released” from inhibition (or transiently depolarized) in order to produce a regenerative response. Our simulations are for a linear array of locally coupled cells, each of Hodgkin-Huxley type with a slowly inactivating T-type calcium current ICa-Tas the PIR mechanism (9). The duration of a rebound depolarization is about 30 to 50 ms, unless cut short by inhibition. Each cell generates a postsynaptic conductance in nearby cells with a decay time constant τsyn of 40 ms and with weights whose spatial footprint is either on-center (Gaussian or square step profile) or off-center (reduced or zero weights near the center of the synaptic footprint).

With on-center inhibition, local stimulation leads to discontinuous recruitment of activity throughout the network (Fig.1A) (10). The activity persists with cells firing synchronously but only in spatially confined clusters, with cluster size of the same order of magnitude as the synaptic footprint's width. Visually, there is some spatiotemporal structure although it is not simple. Clusters do not appear as fixed in space, and successively firing clusters are spatially well separated. The PIR events are brief because of self-inhibition. Therefore cells must receive multiple cycles of brief inhibition in order to adequately prime their PIR mechanism to enable rebound upon release. Consequently, events in a given locale are also sparse in time. The recruitment into quiescent areas proceeds in a lurching manner; after a leading cluster fires, the next downstream firing is delayed by a cycle or more. The discrete progressive aspects show up in diagonal patterning throughout the medium. A cell's membrane potential time course (Fig. 1B) shows growing IPSPs during recruitment before the first rebound spike and highly chaotic fluctuations between spikes, generated completely by deterministic network dynamics. Activity, if summed on appropriate macroscopic spatial scales, looks approximately cyclic in time (Fig. 1, C and D). Target circuits or cells that employ larger convergence factors would receive higher effective frequencies from this network.

Figure 1

Spatiotemporal activity with lurching recruitment waves in a mutually inhibitory network model with on-center coupling. The spatially localized holding stimulus is hyperpolarizing and primes the intracellular PIR mechanism so that PIR occurs at timet = 0, when the stimulus is released. (A) Color plots show membrane potentialV(x,t) (depolarization, red; rest, blue; hyperpolarization, green) of neurons with spatial locationx along the horizontal axis (0 ≤ x ≤ 2 mm) and increasing t (ms) down the vertical axis. The spatial profile of synaptic weights is an on-center Gaussian curve (the curve above the color plot) with footprint width about one-tenth of the network's spatial extent. Default parameter values are used here (9). Initial conditions are as follows:V(x,0) = V rest = −66 mV for each x except in the central region whereV(x,0) = −70 mV for 0.9 ≤ x≤ 1.1; other variables at each location are initialized to their equilibrium values for the specified V(x,0). Time bar, 200 ms; distance bar, 0.25 mm. (B) Membrane potential of a representative neuron. (C) Synaptic activation variable s(t) of a single cell (black trace), averaged over 20% (red trace) or over the whole 100% (blue trace) of the neuronal population, with corresponding power spectra (D). The power spectra were computed by averaging over 10 trials, each with 4096 data points and a time bin of 5 ms.

When synaptic interactions are off-center inhibitory, a different type of recruitment wave is seen, smoothly propagating but still very slow (Fig. 2A) (10). A localized initial stimulus (here, slightly asymmetric) can lead to lurching propagation mediated by cluster-type firing in one direction but smooth nearest neighbor progressive rebounding in the other. Firing durations here are longer than in the on-center case. The smoothly propagating successive cycles lead to periodic signaling across the medium. Such patterns can persist globally (as in Fig. 2A) or, for other parameters or even different initial conditions, the medium can exhibit transient lurching and smooth waves that seem to emerge randomly then disappear (11). The recruitment wave slows if cells experience or need longer hyperpolarization before they can rebound; for example, if either the synaptic time constant τsyn or the time for PIR priming increases (Fig. 2B). When τsyn is too small, synaptic inhibition becomes too brief to elicit postinhibitory rebound excitation, and the wave pattern disappears.

Figure 2

Off-center inhibition leads to both lurching and smoothly propagating recruitment waves. Default parameter values and localized stimulus are as in Fig. 1, except for an additional 2-mV hyperpolarization at t = 0 for 0.86 ≤x ≤ 0.94. Here, coupling is off-center inhibition (profile above the color plot); γ = 1. (A) From each leading cluster (isolated red spots) of the rightward-lurching recruitment wave emerges a smooth wave that travels slowly leftward, with a speed of 0.6 mm/s. The sustained repetitive pattern has a period of 800 ms. Time bar, 200 ms; distance bar, 0.25 mm. (B) The velocity of the leading smoothly propagating wave plotted against the inhibitory synaptic time constant τsyn (left panel) or the scaling factor 1/φ for the cellular time constant τh (right panel) of PIR recovery.

In some parameter regimes, a localized stimulus can initiate propagation of a single smooth PIR pulse (Fig.3B) (10). Solitary waves approaching from opposite directions into resting medium annihilate upon collision (11). Although the behavior appears reminiscent of an isolated action potential propagating along an axon, the mechanism is quite different (12). A major difference is that a PIR wave leaves a trail of synaptic inhibition in its wake. This trailing inhibition can promote a subsequent rebound event as it wanes (as in Fig. 2A). However, in Fig. 3B the time constant for synaptic decay is much smaller than that for repriming the PIR mechanism. The synaptic current dies away too quickly to produce an adequately prolonged inhibition for re-priming (Fig. 3C). In contrast, slower synapses enable the hyperpolarized recovering cells to become extra-permissive, enough to be reexcited after the leading PIR event. The off-center inhibition leads to the double-humped profile of synaptic activity (Fig. 3C).

Figure 3

Dynamic features of local and propagated solitary responses. (A) (left panel) Isolated neuron model shows rebound response upon release from hyperpolarizing current, provided current is adequately strong and of long enough duration. Parameters are adjusted from default values in order to enhance the gradedness of PIR response: g Ca = 0.75 mS/cm2 and τ1 = 0 ms (so that τh = 30 ms); the initial condition is V = −65 mV and h = 0.24 for a holding current of 1 μA/cm2. (Right panel) Representation of an isolated cell's PIR response in the V − h phase plane. Thin solid curves are the V andh nullclines for the resting neuron (21, 22); their intersection (solid circle) represents the stable rest state at (h rest, V rest). For superposed hyperpolarizing input (net input current, −2.5 μA/cm2) the V nullcline moves upward (dashed curve) and there is a new hyperpolarized steady state (open circle); here, h is more permissive than at rest (h = h rest). The thick curve with arrowheads is theV − h trajectory of the PIR response after release of the input current. (B) A solitary smooth wave propagates from a locally stimulated region. The off-center footprint has a square step profile (λ = 200 μm and λgap = 100 μm). Similar results are found for off-center Gaussian synaptic weights. The model with default parameter settings does not support a solitary wave. Here, parameter values that differ from the default are as follows: τ1 = 800 ms, g syn = 2.5 mS/cm2, and kf = 1 (ms)−1, kr = 0.1 (ms)−1. Initial conditions are as in Fig. 1, translated to the right end. (C) Spatial profiles of V,h, and s tot at t* = 1000 ms [thin horizontal line, marked by the asterisk in (B)] for the solitary traveling pulse in (B). The heavy bar indicates 200 ms.

The cellular mechanism that underlies the PIR regenerative response depends on a fast autocatalytic process (in our case, rapid voltage-dependent activation of ICa-T) and a slower negative feedback process (slow inactivation of ICa-T) (13). Suppose that h represents the relative suppressive influence, with h = 0 being maximally negating (at depolarized levels) and h = 1 being permissive (at very hyperpolarized levels); at rest, h = hrest. To prime a neuron for rebounding requires sufficient hyperpolarization for a duration comparable to the time constant τh of the intrinsic PIR mechanism; this allows h to exceed hrest by enough, so that h becomes adequately permissive (Fig. 3A). The neuron is then hyperexcitable and upon release it will rebound. In Fig. 3B, release occurs because of the gap in synaptic drive. Just before rebound, V rises slowly as the gap is first encountered (Fig. 3C). During the depolarized phase, h falls dramatically (its fall terminates the depolarization) to well below hrest.

One can understand the constraints that dictate why PIR wave speed must be slow. In the network, a neuron receives inhibition from its surrounding neighbors weighted according to the synaptic footprint whose characteristic length is λ. If a PIR wave approaches a quiescent cell with speed c , the cell is inhibited for a time approximately λ/ c and this time should exceed τh. Thus, c must be small enough. Moreover, this simple bound (and the computed value of c , see Fig. 2B) decreases inversely with τh. For example, if λ = 100 μm and τh ∼ 100 ms then c must be less than a few millimeters per second, as in the case of lurching waves in the thalamic slice (7, 8). Such speeds are orders of magnitude slower than either the spread of population activity in networks driven by recurrent excitatory coupling (14) or axonal impulse propagation. The priming feature and its influence on speed contrasts substantially with the situation for axonal impulse propagation, where the slower time scales of membrane recovery (like the gating time constant of the delayed rectifier potassium current) hardly influence conduction speed (12).

The possibility that PIR activity may propagate smoothly depends crucially on the interaction between the intracellular (intrinsic) and the intercellular (network) features of our model. If each neuron is rebound-excitable but does not oscillate for maintained input, then the off-center profile of the synaptic weight footprint is important. Without this spatial gap in the inhibitory spread, the essential requirement for space-time translation invariance of a traveling wave cannot be met. Imagine the time course of a neuron during a rebound event. The depolarizing phase is preceded by hyperpolarization and then begins when the cell is released from the GABAA-mediated synaptic input. If this temporal trajectory is to be interchangeable with the spatial profile (as in Fig. 3C), then the locale under a depolarized regime must not be inhibited. This implies no inhibition of self, and by continuity, of nearest neighboring cells, which is an off-center profile for inhibition (15). The narrower this footprint's gap is, the slower is propagation, and if it is too narrow propagation will be precluded.

The substantial differences between the inhibition-induced recruitment that we have shown and the well-known excitation-driven waves may be directly demonstrated with our model. By dynamically modulating the reversal potential V syn of the GABAAsynapses, the synaptic current's sign may be changed, as occurs in the circadian pacemaker network or during development or high activity (16). To illustrate the dramatic effects of such modulation, we slowly increase V syn (from 20 mVbelow resting potential as in Figs. through 3) while locally stimulating the network with brief depolarizing pulses every 500 ms. Using on-center synaptic weights, the activity evolves from spatially localized rebound clustering to quiescence and unresponsiveness whenV syn is near V rest and synapses are truly shunting (Fig. 4) (10). Ultimately, the network gains excitability and the next brief stimulus leads to very rapid propagation of a depolarization wave, with no pre- or posthyperpolarization in the voltage time course. The propagation speed of this event is 25 times faster than that of the smooth PIR wave in Fig. 2A, causing nearly synchronous discharge of the network. A special feature of our rather simple model enables this behavior. At rest, our model's PIR process is partially primed: hrest is not near zero (Fig. 3A). Consequently, the spike generator is not totally disabled. The system is excitable and it responds regeneratively to a brief depolarization. Thus whenV syn exceeds V restsufficiently, this excitability is capable of supporting fast long-range communication via local excitatory but near-shunting interactions.

Figure 4

Change in sign of GABAergic synapses converts slow lurching recruitment by inhibition to nearly instantaneous propagation of the depolarizing event. Parameter values are as in Fig. 1, except that here the on-center footprint has a square step profile. (A) The medium is stimulated periodically in time with a square pulse of current (30 μA/cm2, 50 ms) over the centered region: 0.9 ≤ x ≤ 1.1. The linear ramp ofV syn (t) (15 mV/s) simulates increasing intracellular chloride concentration. Time bar, 500 ms; distance bar, 0.25 mm. The early phase of response shows lurching waves and irregular cluster firings as in Fig. 1A. The medium is inexcitable during the middle phase when V syn is close toV rest; synapses are only shunting. When GABAA synapses are adequately depolarizing from the resting state, recruitment spreads rapidly (∼25 times faster than lurching waves), with near-simultaneous firing in response to the depolarizing stimulus. If V syn were to increase even more, the neurons would steadily depolarize. (B) Magnification of 320-ms time window as indicated by the vertical bar in (A) shows the fast propagation event.

Our finding that localized coupling architecture determines the propagation mode has implications for the thalamic system. Multiple electrode recordings in slices indicate lurching propagation (7) with no reported evidence for smooth waves (although we urge the use of imaging methods whose finer spatial resolution would provide more direct and conclusive evidence). Therefore, based on computations with our idealized model and with a more complete two-layer RE-TC model (17), we suggest that functional coupling in the disynaptic pathway TC-RE-TC is effectively on-center. Consistent with this suggestion, return EPSPs and IPSPs are sometimes seen when individual RE or TC cells, respectively, are stimulated (18). Direct support for autaptic synapses or the footprint shape of localized coupling, although present in some systems (19), is not yet available for the thalamic circuitry. Our results would also apply to an isolated RE subsystem (a thalamic slice in which RE-RE, but not RE-TC-RE, coupling was operative) if experimental conditions were adjusted so that RE cells responded to IPSP barrages with rebound bursts (20).

Our results provoke broader speculation about functional implications for distributed networks of GABAergic PIR neurons. The regularity of the slow smooth wave trains (Fig. 2) suggests utility for a central pattern generator, for example, for driving slow undulatory motion. It is a challenge to account for the emergence of pattern or computational ability on long time scales based on cellular mechanisms with much shorter time scales. Such slow waves might be employed in applications requiring long-delayed dynamic memory. In analogy to how the auditory system identifies the location of sound sources, slow propagation on delay lines could be used for detection of events that are temporally separated by hundreds to thousands of milliseconds. Whether the slow recruitment and maintained inhibition-mediated patterns seen here can still arise in networks having some degree of recurrent excitation will require additional study. Regardless, the different spatiotemporal patterns become signatures for identifying qualitative intrinsic and circuit properties even in isolated networks.

  • * Present address: Center for Neural Science and Courant Institute of Mathematical Sciences, New York University, New York, NY 10003, USA. To whom correspondence should be addressed at the Center for Neural Science, 4 Washington Place, Room 809, New York University, New York, NY 10003, USA. E-mail: rinzel{at}cns.nyu.edu

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