Continuous and lurching traveling pulses in neuronal networks with delay and spatially decaying connectivity

Abstract

Propagation of discharges in cortical and thalamic systems, which is used as a probe for examining network circuitry, is studied by constructing a one-dimensional model of integrate-and-fire neurons that are coupled by excitatory synapses with delay. Each neuron fires only one spike. The velocity and stability of propagating continuous pulses are calculated analytically. Above a certain critical value of the constant delay, these pulses lose stability. Instead, lurching pulses propagate with discontinuous and periodic spatio-temporal characteristics. The parameter regime for which lurching occurs is strongly affected by the footprint (connectivity) shape; bistability may occur with a square footprint shape but not with an exponential footprint shape. For strong synaptic coupling, the velocity of both continuous and lurching pulses increases logarithmically with the synaptic coupling strength g_syn for an exponential footprint shape, and it is bounded for a step footprint shape. We conclude that the differences in velocity and shape between the front of thalamic spindle waves in vitro and cortical paroxysmal discharges stem from their different effective delay; in thalamic networks, large effective delay between inhibitory neurons arises from their effective interaction via the excitatory cells which display postinhibitory rebound.

Neuronal population discharges appear in disinhibited coronal neocortical slices in response to electrical stimulation above a certain threshold. These slice preparations were developed initially as experimental models for epilepsy (1, 2). Recently, experimental and theoretical investigations (3–7) have tried to relate the dynamics of propagating discharge to the underlying neuronal circuitry, and to use the dynamics of cortical slices as a first stage toward understanding spatio-temporal dynamics in neuronal networks (8–10). The average discharge velocity is about 10–15 cm/s (3); neurons are recruited to the wave because of the excitatory recurrent interactions between neurons. Numerical simulations of a conductance based neuronal model with homogeneous architecture reveal that the discharge propagates at a constant velocity, as a continuous traveling pulse (3, 6, 7). Spatial inhomogeneities in the velocity, which were discovered experimentally, were attributed to spatial fluctuations in the synaptic and intrinsic neuronal properties along the slice (11, 12). In both theory and experiment, there was a minimal velocity below which the discharge could not propagate. Propagating discharges with similar properties and velocities have been found in other cortical structures, such as the hippocampus (13, 14) and the piriform cortex (15).

Another example of propagating discharges is the spindle-like discharges in thalamic slices (16, 17). The circuit is composed of excitatory thalamocortical (TC) cells and inhibitory reticular (RE) thalamic cells, coupled with reciprocal synaptic connections. The excitatory cells possess a postinhibitory rebound mechanism (see, e.g., ref. 18). The propagation velocity is around 1 mm/s (19). Numerical simulations have indicated that these discharges propagate in a nonsmooth, periodic, “lurching” manner (20, 21). Each recruitment cycle has two stages. At the first stage, a new group of inhibitory RE cells is excited by synapses from TC cells, and this RE group inhibits a new group of silent TC cells. At the second stage, the newly recruited TC cells rebound from hyperpolarization and fire a burst of action potentials. These bursts further recruit more RE cells during the next cycle. Thus, the lurching wave has a periodic nature at the front. Rinzel et al. (22) have reduced the model of a thalamic slice to a model of coupled inhibitory units and found that the model can also exhibit lurching waves if the synaptic reversal potential is hyperpolarized enough. With off-center (coupling that is zero at zero distance and has a maximum at a finite distance) inhibition, lurching waves can propagate in one direction while continuous waves can propagate in the other direction, demonstrating bistability§ in the system. An experimental in vivo study suggested that the RE nucleus may generate spindle-like oscillations (23). Numerical simulations of isolated RE networks showed that such oscillations can propagate as lurching pulses (20) or as continuous pulses (24).

In this work, we explore the recruitment stage of these two propagating discharges. We ask the following questions.

1.

What is the basis for the different types (continuous or lurching) of propagating discharges? How is the type determined by the network architecture, the kinetics of single cells and synapses, and the synaptic delay?

2.

What is the relationship between the velocity of the pulse (or wave), and the intrinsic, synaptic, and architecture properties of the system?

Specifically, we want to know why cortical networks exhibit fast, continuous discharges, whereas thalamic networks exhibit slow, lurching discharges. To answer these questions, we develop a joint conceptual framework for the two types of networks, which enables us to compare the two preparations and the two types of discharge propagation. We show that under certain conditions and approximations, the discharge dynamics of both cortical and thalamic slices can be reduced to a model of integrate and fire neurons in which each neuron can fire only one spike. Neurons are coupled by excitatory synapses with delay, which is small for cortical networks and large for thalamic networks. Using analytical and numerical methods, we find that continuous pulses exist and are stable for small values of constant (space-independent) delay. As the constant delay increases, this pulse loses stability and lurching pulses are obtained.

The Model

Reduction of Models of Paroxysmal Discharges in Disinhibited Cortical Networks.

Simulations of a conductance-based models of paroxysmal discharges in disinhibited cortical tissues (Fig. 1A) reveal that the discharge velocity ν is determined primarily by the response of the postsynaptic neuron to the first one or two spikes of the presynaptic neuron (3), especially with prominent synaptic depression. Therefore, taking only the first spike of each presynaptic neuron into account and ignoring all the others is a good approximation (6). For example, if we assume that only the first spike elicits an excitatory postsynaptic conductance (EPSC) for the parameters of figure 8 in ref. 3, the velocity decreases by only 15%.

Figure 1.

(A) Synaptic architecture of the cortical and thalamic models. Both have a one-dimensional architecture, with the coupling between cells decaying with their distance. The footprint lengths are denoted by σ (3, 20). In the case of two populations, the first and second letters in the subscript denote the pre- and postsynaptic populations respectively. (A) Disinhibited cortical model: a chain of excitatory (E) cells. (B) Thalamic model. Inhibitory (I) RE cells inhibit excitatory (E) TC cells, and TC cells excite RE cells. Mutual inhibition between RE cells is neglected because it has relatively small effect on discharge propagation (16, 19). (C) Exponential (solid line) and square (dashed line) footprint shapes.

Even adjacent cortical neurons have a delay of about 2 ms (25, 26); more distant neurons are expected to have a larger delay because of the finite axonal connectivity. The delay τ_delay between neurons at positions x and x′ is therefore

1

where τ_d is the constant delay and c is the axonal conduction velocity.

Reduction of Models of Spindle-Like Discharges in Thalamic Networks.

In this paper, we consider a spatially structured network of excitatory neurons, which may be interpreted as a reduction from a two-population thalamic network (Fig. 1B). The idea is that because the RE-to-TC projection is topographic and acts via GABA_A and GABA_B receptors, excitation of one RE cell would result in a delayed barrage of EPSCs in the neighboring RE cells through the disynaptic RE-TC-RE loop. In this idealized view of the isolated thalamic circuit, the RE cell layer acts effectively as a cell population with reciprocal AMPA-mediated excitatory interactions, with an effective delay τ_d (Eq. 1) of order 100 ms caused by the time needed for the TC cell to rebound from inhibition. Because we are interested in this work only in the recruitment process, we can model the system by considering only the first “spike” each cell fires. The large effective delay ensures that the velocity at moderate coupling is determined mainly by the first spike. This spike represents a Ca²⁺ spike upon which rides a train of action potentials (20). The assumption of fixed effective delay only applies if those TC cells that receive inhibitory input are in burst-capable mode.

Model Description.

We use the integrate-and-fire model (6, 27)

2

for 0 < V(x, t) < V_T, where V(x, t) is the membrane potential of a neuron at a position x and time t, τ₀ is the passive membrane time constant of the neuron, I_syn is the normalized synaptic input, and I_app is the normalized applied current; I_app = 0 unless stated differently. When V of a neuron reaches the threshold V_T at time T(x), the neuron fires a spike, and cannot fire more spikes afterwards; its voltage is set to zero at later times. We assume that the number of neurons within a footprint length is large and therefore use a continuum model and replace the sum over the presynaptic neurons by an integral:

3

where g_syn = g̃_synΔ/C, g̃_syn is the synaptic conductance, C is the membrane capacitance, and Δ = V − V_syn is approximated here to be a constant (“coupling by currents”, see, e.g., refs. 6 and 28); τ_delay is the delay between the post- and presynaptic neurons. The temporal shape of the EPSC that a postsynaptic cell at a position x receives following a spike of a presynaptic cell at a position x′ is given by the normalized α function α[t − T(x′)]:

4

where τ₁ and τ₂ are the synaptic rise and decay time, respectively. The spatial dependence of the synaptic strength on the distance between neurons, w(x), is called the “synaptic footprint shape” (3, 20). We examine two shapes (Fig. 1C):

5

6

σ is called the “synaptic footprint length.”

Define the response (Green’s) function G(t) for t > 0 as

7

and G(t) = 0 for t < 0. Then, for τ₁ = 0,

8

for t > 0. The function G is the normalized EPSP (excitatory postsynaptic potential) developed in the cell as a response to the EPSC (Eq. 4). The Volterra representation of Eqs. 2 and 3 for neurons that can fire only one spike is

9

together with the condition that T(x) is the first time that the voltage crosses the threshold. This condition requires that

10

Numerical Methods.

Eqs. 2 and 3 are simulated numerically by discretizing space. There are N neurons in the chain, and the density of neurons is ρ per length σ. The coupled system of ordinary differential equations for the integrate-and-fire neurons is solved by using exact integration (29). To stimulate the network, applied current is “injected” into a group of neurons on the “left” of the system (small x values) that span a length at least equal to the footprint length σ (“shock” initial conditions).

Results

In the following, we calculate the existence and stability regimes and the velocity of the continuous and lurching pulses for c → ∞ (infinite axonal conduction velocity). The effect of finite axonal velocity c is reported at the end of this section. In most calculations we will take τ₁ = 0 for simplicity, unless stated otherwise. Details of the analytical calculations will be published elsewhere.

Continuous and Lurching Pulses.

A pulse can propagate along the network in response to “shock” initial conditions. For zero or small τ_d (below a critical value τ_dc), the pulse is continuous far from the stimulus region (Fig. 2A), and the firing times of the neurons obey T(x) = T₀ + x/ν, where ν is the pulse velocity and T₀ is an arbitrary time. The neuronal potential satisfies a traveling pulse equation as well: V(x, t) = Ṽ(x − νt) (3, 6, 7). For large τ_d (above τ_dc), a lurching propagating pulse is observed (Fig. 2B). Space is spontaneously segregated into basic spatial units, each with a spatial period length L, and the firing time in each unit can be obtained from the spatial period in the previous unit according to

11

The average velocity of the pulse is ν = L/T_per. One can easily show that the function T(x) − x/ν is a periodic function with a period L, as demonstrated in Fig. 2C.

Figure 2.

(A and B) Rastergrams obtained from simulating Eqs. 2, 3, and 5 with the condition that each neuron can fire only one spike. Parameters: τ₀ = 30 ms, τ₂ = 2 ms, g_synV_T = 10, c → ∞, N = 5 × 10⁴, ρ = 500; for these parameters, τ_dc = 11.15 ms. The solid circles represent the firing time of neurons as a function of their normalized position x/σ; spikes of only one out of every 50 neuron are plotted. Together, the groups of solid circles looks almost like one continuous line. (A) For τ_d < τ_dc (3 ms), a continuous pulse is obtained. (B) For τ_d > τ_dc (30 ms), the pulse is lurching. (C) The fluctuation around the constant-velocity solutions, T(x) − x/ν are plotted as a function of position x/σ for the same parameters as in B, to demonstrate the oscillatory nature of the lurching pulse. The firing times of all the neurons are plotted here as solid dots.

Existence, Stability, and Velocity of Continuous Pulses.

General formalism.

Substituting the condition for a continuous pulse, T(x) = x/ν, into the evolution Eq. 9, we obtain

12

together with the condition (a consequence of Eq. 10)

13

Stability of the continuous pulse is calculated by analyzing spatial perturbation of Eq. 9 (see also ref. 7).

Exponential footprint shape.

The velocity ν is determined using Eqs. 5, 8, and 12:

14

together with Eq. 13; this is an extension of the equation obtained in ref. 6 for τ_d = 0. From this equation, one can see the following:

1.

For τ_d = 0, the left-hand-side of Eq. 14 has a minimum with respect to ν at ν_min = σ/. Continuous pulses cannot propagate¶ with velocity smaller than ν_min, which is obtained for a minimal synaptic coupling g_syn,min.

2.

Because exp(τ_dν/σ) > 1 and increases with τ_d, ν_min decreases with τ_d and is obtained for larger g_syn,min.

3.

For τ_d > 0 and at large enough g_syn, the velocity is determined mainly by the exponential factor in Eq. 14 and therefore depends logarithmically on g_syn to the highest order. In contrast, for τ_d = 0, the velocity exhibits a power-law dependence on g_syn at large g_syn (6).

Graphs of ν as a function of 1/g_syn for several values of τ_d, together with results of the stability analysis of the pulses, are shown in Fig. 3. For large enough τ_d, the pulse is unstable for small g_syn (and low velocities). The various behavioral regimes of the continuous pulse in the τ_d − g_syn plane are plotted in Fig. 4A. Below the solid line, the continuous pulse does not exist. Above this line, the continuous pulse is stable if the delay τ_d is smaller than a critical delay τ_dc and is unstable otherwise. The dashed line denotes the value of τ_dc as a function of g_syn. At very large g_syn, τ_dc increases logarithmically with g_syn.

Figure 3.

The velocity of the continuous pulse as a function of V_T/g_syn for several values of τ_d and exponential footprint shape (Eq. 14). The wide lines represent stable pulses and the narrow lines represent unstable pulses. The number above each line, from 0 to 50, denotes the value of τ_d. Parameters: τ₀ = 30 ms, τ₂ = 2 ms.

Figure 4.

Regimes of existence and stability of the continuous and lurching pulses in the τ_d-ν plane are shown in A for exponential footprint shape and in B for square footprint shape. Parameters are as in Fig 3. The boundaries of the regime in which the lurching pulse exists and is stable were computed from numerical simulations, in which a pulse was initiated by a “shock” initial stimulus; N = 20,000, ρ = 50. The solid line denotes the minimal possible velocity as a function of g_syn; the continuous pulse becomes unstable (via a Hopf bifurcation) on the dashed line. The continuous pulse is therefore stable above both the solid and long dashed line, as denoted by “s.” It is unstable between the two lines, as denoted by “us,” and does not exist below the continuous line, as denoted by “ne.” The light-gray shading represents the region for which lurching pulses (and not continuous pulses) are obtained. Bistable regimes, in which the continuous pulse can coexist with the lurching pulse, are denoted by the dark-gray shading. For the square footprint shape (B), but not for the exponential footprint shape, there is such a bistable regime that has a “tongue-like” structure. The arrow at the right of each graph represents the minimal values of g_syn for which the lurching pulse is found in simulations for τ_d → ∞.

The boundaries of the regime of existence of lurching pulses have been located by using numerical simulations, and they are represented by the light-shaded area in Fig. 4A. Lurching pulses are observed in all the parameter regimes in which the continuous pulse is unstable. In addition, lurching pulses are also observed in a parameter regime in which the continuous pulse does not exist at all. At very large delay, lurching pulses are observed above a critical value of g_syn, denoted by the arrow, that does not depend on τ_d (see below).

Square footprint shape.

The velocity ν is determined by using Eqs. 6, 8, and 12:

15

The qualitative results regarding the minimal velocity and its dependence on τ_d are the same as for the exponential case. The situation is different, however, for large g_syn. For ν = σ/τ_d, the right-hand side of Eq. 15 is zero. Hence, at the limit g_syn → ∞ the velocity of the continuous pulse approaches the finite value σ/τ_d.

The different behavioral regimes of the continuous pulse for a square footprint shape are presented in Fig. 4B. The critical delay τ_dc increases with g_syn almost linearly at large g_syn. As a result, for a specific τ_d there is a moderate g_syn value for which the continuous pulse is stable. Lurching pulses are obtained in the region denoted by the gray shading in Fig. 4B. There are two apparent differences between the situation here and the situation for exponential footprint shape. First, lurching pulses exist in an area which is composed of “tongues.” Second, a bistable regime exists, denoted by the dark-gray shading, in which the two types of pulses can propagate, depending on the initial stimulation (22).

Velocity of Lurching Pulses.

How does the velocity of the lurching pulse depend on the synaptic strength? We can calculate this velocity in the case

16

This case corresponds to a large delay and fast EPSCs (as in thalamic slices); note that τ₁ = 0. Simulations of such cases show that neurons fire only during a time period that is small in comparison to the delay, and L is almost unaffected by the delay period as long as it is large enough. Therefore, the pulse velocity is ν = L/τ_d. For τ_d ≫ τ₀, a neuron that fires during the nth lurching period (with length L and time T_per) is affected only by neurons that have fired during the previous lurching period. The contribution to the potential of that neuron of neuronal EPSCs from neurons in earlier periods has already decayed, mostly because of the large delay and also because of the fact that neuron in earlier lurching periods are more distant from that neuron. The neuron is also not affected by neurons that fire during the same lurching period. We assume that the lurching wave is initiated at very large, negative x, and a lurching spatial period starts at x = 0. We define the function f(x) on the interval 0 < x ≤ L to be f(x) = T(x + nL) − nT_per, where n is the integer part of T(x)/T_per (or x/L). Neurons at a position 0 ≤ x < L will be affected only by neurons located in the interval −L ≤ x < 0. Eq. 9 becomes

17

where T̂ = T_per − τ_d. This means that for a square footprint shape, L ≤ σ. Using an asymptotic expansion, we find that the length L for an exponential footprint shape is given, for the conditions in Eq. 16, by

18

For large g_syn, expanding this equation yields L = σ ln[g_syn/(2V_T)]. Eq. 18 has a solution only if g_syn > 8V_T, and this is the synaptic conductance threshold for enabling the propagation of lurching waves under the conditions of Eq. 16. The dependence of the lurching spatial period L (in units of σ) as a function of g_syn/V_T is shown in Fig. 5. The straight line in logarithmic scale shows that except for g_syn near the threshold, L increases logarithmically with g_syn. The open circles represent simulation results with τ_d = 1,000 ms, τ₀ = 30 ms, and τ₂ = 0.002 ms, and they fall exactly on the analytical curve.

Figure 5.

The normalized length of the lurching period L/σ as a function of g_syn/V_T. The analytical solution for the case τ₁ = 0, τ₂ ≪ τ₀ ≪ τ_d (Eq. 18) is represented by the solid line. Simulations were carried out with N = 200,000 and ρ = 500. Simulations results with the corresponding parameter set: τ_d = 1000 ms, τ₀ = 30 ms, τ₁ = 0, τ₂ = 0.002 ms, denoted by ○, fit the analytical solutions almost exactly. The symbol × denotes simulations with τ_d = 20 ms, τ₀ = 30 ms, τ₁ = 0, τ₂ = 0.002 ms; the symbol ∗ denotes simulations with τ_d = 20 ms, τ₀ = 30 ms, τ₁ = 0, τ₂ = 2 ms; and the symbol □ denotes simulations with τ_d = 20 ms, τ₀ = 30 ms, τ₁ = 0.3, τ₂ = 2 ms.

Using simulations, we tested the validity of the perturbation calculation when the time constants of the system do not fulfill Eq. 16 (not near the threshold). First, we reduced τ_d to 20 ms. This change has almost no effect for large g_syn and mildly increases L for small g_syn. This velocity increase can be attributed to the excitatory effect on a neuron from neurons in cycles before the immediate previous lurching cycle; it is stronger for small g_syn because of the shorter L. Second, we increased τ₂ to 2 ms. As a result, L decreases, because the EPSP developed in the postsynaptic cell (Eq. 7) is smaller as a result of the interplay between the EPSC and the leaky neuronal integrator. This change of τ₂ has, however, an effect of less then 10% on L in comparison to the analytic result (Eq. 18). Third, we simulated a network with τ₁ = 0.3 ms. This change did not change L significantly in comparison to the case τ₁ = 0. These results show that our analytical theory yields a good approximation even beyond the parameter regime for which it is derived.

Finite Axonal Velocity.

When the axonal velocity c is finite, we find that the pulse velocity ν is given by the equation 1/ν = 1/ν_∞ + 1/c, where ν_∞ is the pulse velocity for infinite axonal velocity. The value of τ_dc for a specific value of g_syn, and therefore the parameter regime in which stable continuous pulses are observed, do not depend on c.

Discussion

The main results of this work are as follows: (i) Continuous pulses can propagate along one-dimensional neuronal networks with small constant delay τ_d, and lurching pulses with periodic characteristics can propagate with large τ_d. (ii) The functional dependence of the propagation velocity ν on the synaptic coupling strength g_syn is hardly affected by the pulse type; velocities of both continuous and lurching pulses increase logarithmically with the g_syn for a exponential footprint shape; they are bounded for square footprint shape (this is consistent with the heuristic argument of ref. 20). (iii) The footprint shape strongly affects the types of pulses that are obtained with intermediate τ_d and g_syn. In particular, bistability (a regime where both pulse types propagate) can occur with square footprint shape [or with an off-center shape (22)] but not with exponential shape. From these results we conclude that the difference in form and velocity between cortical and thalamic propagating discharges stems from the much larger effective delay in thalamic networks.

The instability of continuous pulses with large τ_d has been shown independently by Bressloff (7). We find that lurching pulses are obtained at these value of τ_d, and sometimes they even coexist with the continuous pulses. Traub and colleagues (13, 14) have numerically studied the velocity reduction due to finite c. We find a simple analytical formula that relates the velocity with finite c to the velocity in the limit c → ∞. A heuristic argument showing that the velocity of the lurching pulse increases logarithmically with g_syn at large g_syn was presented in ref. 20 (but see also ref. 30). Here, we derive the exact relationship between the velocity and g_syn in the limit τ₂ ≪ τ₀ ≪ τ_d, and show numerically this this result is approximately valid even well beyond this parameter regime.

In our model, we are interested in the recruitment processing of neurons into the activity. We do not deal with the period of firing and its termination. Each neuron is allowed to fire only one spike. This simplification, which allows analytical treatment of the model, is justified for cortical tissues, in particular with prominent synaptic depression, because the velocity is determined primarily by the response to the first presynaptic spike. It is justified for the thalamic network because of the large effective delay. At g_syn values near the threshold for lurching, however, subsequent spikes may have an effects. As a result, very low velocities can be obtained in simulations of thalamic networks (20), whereas in our model velocity cannot be smaller than a finite minimal velocity.

In this work, the thalamic network model is reduced to a network of reticular thalamic neurons coupled with effective excitation with delay. Rinzel et al. (22) have reduced the same network into a system of thalamocortical cells coupled by effective inhibition. The two pictures are complementary. Our reduction is exact in the limit σ_IE ≪ σ_EI (Fig. 1B), where σ_IE and σ_EI denote the inhibitory-to-excitatory and the excitatory-to-inhibitory footprint lengths, respectively, whereas the reduction of ref. 22 is exact in the limit σ_IE ≫ σ_EI. Experimental results and computational models indicate that the sum of these two lengths is small, of order 100 μm (20, 21), but there is yet no direct measure of each length separately (31, 32). Rinzel et al. (22) showed that propagation can proceed smoothly with off-center footprint shape in a parameter regime in which lurching occurs with an on-center footprint shape; in some parameter regime, the wave can lurch in one direction and propagate smoothly in the other direction, exhibiting bistability. Here we show that the parameter regime in which lurching or smooth propagation occur depends on the footprint shape. Preliminary simulations of the corresponding RE-TC network model (20) indicate that for a square footprint range there is a transition from a lurching wave to a continuous wave as the AMPA conductance increases, with possible bistability, consistent with the results of the present model; “tongue”-like structures, however, were not observed.

Our model constitutes a framework for comparing and explaining the velocity and type of pulses and waves in various tissues, whose dynamics can be reduced to that of Eqs. 2 and 3. The main reason that thalamic spindle-like waves propagate much more slowly than paroxysmal discharges in neocortex, hippocampus, and piriform cortex is that the effective delay in the thalamic network is much larger (≈100 ms in comparison to ≈2 ms in cortex). When inhibition is mediated by GABA_B inhibition only, the time needed for a TC neuron to rebound from hyperpolarization is larger in comparison to the case when GABA_A inhibition is intact; the number of Na⁺ action potentials within a burst is more prolonged (16, 17), which may result in a stronger effective g_syn. The combined effect is that the velocity is reduced with GABA_A blockade (19–21). Paroxysmal discharges propagate in disinhibited tangential slices of neocortical layer IV with a velocity which is about an order of magnitude smaller than the propagation velocity in disinhibited coronal cortical slices (33). The slow velocity is partly explained by the fact that these discharges are mediated by slow NMDA receptors (larger τ₁ and τ₂, see Eqs. 4 and 14), but assuming that the footprint length in layer IV is small in comparison to that in layer V [which mediates mostly the propagation in coronal slices (34)] is also needed to explain the slow velocity. Measurements of the minimal velocity ν_min [by gradual blockade of excitatory EPSCs (3)], together with our theoretical model, should be carried out in order to estimate the footprint length in layer IV networks. The model predicts that if the effective synaptic delay or conductance strength could be manipulated experimentally, there may be a transition from continuous to lurching pulses. The values of the parameters at the transition depend on the foot-print shape.

Our theory enables us to estimate the footprint length based on the critical velocity, provided that we know the footprint shape, the synaptic kinetics, and the delays τ_d and c. Experiments on synaptic plasticity reveal that the response of a neuron to the first presynaptic spike may vary (either be enhanced or be depressed) in response to artificial (35) and natural (36, 37) conditions. If indeed the global excitation g_syn is increased but the footprint shape remains unchanged, at least within a restricted domain, the discharge velocity in a disinhibited slice should increase but the minimal velocity ν_min remains unchanged. Therefore, performing such experiments can corroborate or refute theories about global synaptic enhancement or reduction.

Abbreviations

TC

thalamocortical

RE

reticular

EPSC

excitatory postsynaptic conductance