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1Center for Theoretical Biological Physics; 2Department of Physics; 3Graduate Program in Neurosciences, University of California San Diego, La Jolla, California; 4Department of Physiology and Zlotowski Center for Neuroscience, Ben Gurion University of the Negev, Beer-Sheva; and 5Department of Neurobiology, The Weizmann Institute of Science, Rehovot, Israel
Submitted 12 July 2005; accepted in final form 26 October 2005
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ABSTRACT |
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INTRODUCTION |
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), auditory (Suga et al. 1997; Zhang and Suga 1997), and tactual (Castro-Alamancos 2002b; Diamond et al. 1992; Nicolelis and Chapin 1994; Richardson 1973; Temereanca and Simons 2004) thalamic nuclei do not act as a relay of sensory input but rather process sensory information (Guillery and Sherman 2002a). In the rodent vibrissa somatosensory system, input from brain stem nuclei is transformed by three interacting thalamic nuclei (Fig. 1A); the posterior medial (POm) thalamic nucleus, the ventroposterior medial (VPm) thalamic nucleus, and the reticular (Rt) thalamic nucleus (for a review of anatomy, see Deschenes et al. 1998; Diamond 1995; Kleinfeld et al. 1999). The POm and VPm nuclei receive afferent input primarily from trigeminal nucleus principalis (Pr5) and spinal nucleus interpolaris (Sp5I) (Veinante et al. 2000) and project reciprocally to the same area of the Rt nucleus (Crabtree and Isaac 2002; Crabtree et al. 1998). The Rt nucleus, in turn, provides feedback inhibition to the thalamocortical neurons in both POm and VPm nuclei via fast GABAA- and slow GABAB-mediated synaptic currents (Von Krosigk et al. 1993). The possible mixing of signals between POm and VPm nuclei by Rt neurons forces consideration of the full POm-Rt-VPm circuit as a means to delineate the thalamic dynamics.
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, 2001; Sosnik et al. 2001). The time course of the motion of the vibrissae during whisking, albeit not the mechanics (Szwed et al. 2003), was approximated by the application of periodic air puffs to one or two rows of vibrissae. The latency and rate of spiking for brain stem units were observed to be independent of the stimulus frequency for frequencies between 2 and 11 Hz (Ahissar et al. 2000; Deschenes et al. 2003; Jones et al. 2004; Sosnik et al. 2001), which incorporates the range of natural exploratory whisking (Berg and Kleinfeld 2003; Welker 1964). While the response latency of units recorded in VPm thalamus was also nearly independent of stimulus frequency (Ahissar et al. 2000; Hartings and Simons 1998; Hartings et al. 2003; Sosnik et al. 2001), the response latency of units recorded from the POm thalamus increased considerably with frequency for stimuli with a 50-ms, albeit not a 20-ms rise time (2000, Ahissar et al. 2001; Sosnik et al. 2001). Thus POm but not VPm thalamus can code stimulus frequency in terms of latency. In addition, there is a clear coding of stimulus frequency in the rate of VPM neurons.
We consider the hypothesis that the essential determinants of the latency coding observed in POm thalamus depend on both features of the internal circuitry, i.e., the strong facilitation as well as delayed and prolonged response of GABAB-mediated synaptic transmission (Kim et al. 1997), and features of the external stimulus, i.e., the gradually increasing spike rate of the brain stem input to thalamic nuclei. Our challenge is to address this hypothesis in terms of a neural-based model and account for the differences in the response of neurons in POm versus VPm thalamus to periodic stimulation. We ask: why does the latency in the response of POm neurons to prolonged stimuli increase substantially with frequency during steady state conditions? Why is there a distinct difference in latency coding for POm versus VPm thalamus? And why is there no latency coding for brief stimuli? A distinctive feature of our approach is the use of reduced models that are amenable to analytical treatment as a means to gain intuition into the role of synaptic versus external inputs as well to gain insight into the stability and sensitivity of solutions of the model.
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MODEL |
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The architecture of the model includes the three thalamic nuclei: POm, VPm, and Rt. The Rt nucleus receives fast excitatory input from both POm and VPm nuclei. In turn, the POm and VPm nuclei receive inhibitory feedback from Rt thalamus as a fast GABAA- and delayed and prolonged GABAB-mediated current (Fig. 1A). As a means to obtain insight into the neuronal dynamics and to develop a set of robust predictions, we examine a succession of models of increasing complexity. We begin with a reduced but analytically tractable circuit comprising only the POm and the Rt nuclei (Fig. 1B). Then, we investigate the effects of feedforward input from the VPm to the Rt (Fig. 1C). Finally, we study the full thalamic architecture that further includes feedback inhibition from the Rt to the VPm nucleus (Fig. 1D).
We note that the difference in latency coding between the VPm and POm thalamus is also reflected by neurons in cortical layers that receive axonal projection from these nuclei, i.e., layers IV and Va, respectively (Ahissar et al. 2000, 2001; Ahrens et al. 2002). We focus on feedback dynamics at the level of the thalamus. We test if GABAB-mediated currents and the brain stem stimulus shape are sufficient to explain the observed differences in latency.
Measured spike rates
The measured peristimulus time histograms (PSTHs) in brain stem nuclei Pr5 and SpVI and in the thalamic nuclei VPm and POm, based on population-average data from all the recorded multiunits (Ahissar et al. 2000, 2001; Sosnik et al. 2001), show different steady-state responses to periodic air-puff stimuli, each 50 ms in duration, delivered at frequencies of fstim = 2, 5, 8, and 11 Hz (Fig. 1A). The population-averaged temporal responses for both the Pr5 and the Sp5I nuclei of the brain stem may be described as a dual-sloped ramp in which a brief, rapidly rising phase is followed by a prolonged, slowly rising phase. The neuronal activity then gradually decays after the activity reaches its maximum value. In contrast to the prompt response in brain stem, the activity of units in POm thalamus are delayed and then rise to their maximal value. Critically, the response rises more gradually to its peak value as the stimulus frequency, fstim, is increased (cf. 2- and 11-Hz data in Fig. 1A), so that the rise time is, very roughly, inversely proportional to fstim, whereas the maximal activity decreases with fstim. In contrast to the case for POm thalamus, the activity of units in VPm thalamus begins with shorter delay and then rises rapidly to a peak. The rise time is almost independent of fstim, but the maximal activity decreases with fstim.
Rate model
Our technical approach makes use of rate equations (Hopfield 1982; Wilson and Cowan 1973), where the instantaneous spike rate of a population of neurons is expressed in terms of the presynaptic activation for each synaptic current (Ermentrout 1994; Kang et al. 2003; Rinzel and Frankel 1992; Shriki et al. 2003). The activation parameters are denoted by subscripted ui(t), where the subscript i is used to denote the presynaptic neuronal population. For inhibitory conductances, a second subscript, A or B, denotes whether the synapse is mediated by GABAA or by GABAB, respectively. Thus as examples, uPOm(t) describes activation of the excitatory current that originated from POm neurons and uRt,B(t) describes activation of the slow inhibitory current that originates from the Rt nucleus. We further note the aggregate measure of the strength of the synaptic connection of the projection from nucleus j to nucleus i by the synaptic conductance gi;j. For example, gPOm;Rt,B denotes the GABAB-mediated input from Rt to POm thalamus. The activation parameters ui(t), multiplied by the synaptic conductances gi;j, determine the total synaptic current to the postsynaptic cell, e.g., gPOm;Rt,BuRt,B(t) for GABAB-mediated input to POm neurons from Rt thalamus.
NONLINEAR, DELAY DIFFERENTIAL EQUATIONS THAT DEFINE THE DYNAMICS.
The observable quantities in a network are the instantaneous spike rates for each population of neurons. This rate is denoted by Mi for nucleus i, e.g., MPOm for the POm nucleus. Without intrinsic adaptation, the presynaptic spiking rate Mi is approximated by the instantaneous input-output relation, or f-I curve, Mi(t) = [I(t)]+, and is taken to be the rectification (linear-threshold) function, that is
 | (1) |
The total current I(t) is the sum of the synaptic currents and the brain stem input, i.e., IPOm(t) and IVPm(t) for the POm and VPm nuclei, respectively, relative to the threshold for spiking, e.g., 
POm for neurons in POm thalamus. Thus the instantaneous spiking rates of the three thalamic nuclei are
 | (2) |
 | (3) |
 | (4) |
) defined through the dynamics of the variables aVPm and aPOm (see following text). All the variables and parameters, except for time, are normalized and are unit-less in our formulation (Ermentrout 1994; Kang et al. 2003; Shriki et al. 2003). Equations 2–4, with nonnegative values of the conductances, correspond to the architecture of Fig. 1D.
The dynamics for glutamatergic fast excitatory synapses and GABAA-mediated inhibitory synapses, but not GABAB-mediated inhibitory synapses, are defined by pairs of delay differential equations
 | (5) |
 | (6) |
 | (7) |
 | (8) |
G, 
G, and tG are the synaptic rise, decay, and delay times, respectively, of the excitatory synapses, A, A, and tA are the synaptic rise, decay, and delay times, respectively, of the GABAA-mediated synapses, the variable xi is an auxiliary variable, and the subscript i is POm thalamus or VPm thalamus. The synaptic delays depend only on the synaptic phenotype in our architecture.
Inhibitory synapses mediated by GABAB are nonlinear and facilitating. In practice, they have a delay of 
30–40 ms and respond much more strongly to a prolonged burst of spikes than to a brief burst (Golomb et al. 1996; Kim et al. 1997). We use a version of a nonlinear model for GABAB-mediated synapses (Golomb et al. 1996)
 | (9) |
 | (10) |
B, B, and tB are the synaptic rise, decay, and delay times, respectively, of the GABAB-mediated synapses. The quadratic nonlinearity in the first term in the right-hand side of Eq. 10 is responsible for the facilitating nature of the synapse.
ADAPTATION.
We incorporate an idealization of cellular adaptation in the rate equations for the VPm and POm thalamic nuclei with an adaptation time scale that is much slower than the spiking time scale. A thalamic neuron fires a small number of spikes in response to a single stimulus in the relay mode (Minnery and Simons 2003; Sosnik et al. 2001). In particular, cells in POm thalamus rarely fire more than one spike in response to a single stimulus (Sosnik et al. 2001). We therefore model adaptation as a process of inactivation and slow recovery of the "neuronal pool" that is capable of spiking (Eggert and van Hemmen 2000, 2001). Such a process can be described as a multiplicative dynamical process with two time scales, one of activation as a result of neuronal activity and one of inactivation when neurons are silent. The equations for the adaptation variables ai (Eqs. 2 and 3) and the auxiliary variables bi, identified with the activation dynamics, are
 | (11) |
 | (12) |
We assume that POm thalamus has stronger adaptation than VPm thalamus; this allows us to model the fast decay of POm activity at low frequencies in comparison to the more sustained activity of VPM neurons (Fig. 1A). The adaptation in the VPm neurons rises faster than that in POm neurons as a result of the higher spiking rate in the former nucleus. The activity in the Rt nucleus is more prolonged than the activity in thalamic relay nuclei (Hartings and Simons 2000), and therefore we do not introduce adaptation for the Rt nucleus.
STIMULUS SHAPE.
The input from the brain stem nuclei is considered as an external variable that monotonically tracks the stimulus. We do not discriminate between inputs from the Pr5 and SP5I nuclei as the average responses of neurons to air-puff stimuli are very similar in the two areas (Ahissar et al. 2000). In both dorsal thalamic nuclei, the input from the brain stem begins after a short delay that represents the onset of the stimulus and the time that the stimulus drives the thalamic nuclei. We model the "double ramp" shape of the brain stem input to VPm thalamus, IVPm, as a piecewise linear function. The brain stem input to POm thalamus is proportional to the brain stem input to the VPm, and it is delayed to allow larger latency in POm even for low stimulus frequencies
 | (13) |
The parameters of the brain stem stimuli to the two thalamic nuclei are given in APPENDIX A.
PARAMETERS.
We use the following parameters throughout the analysis unless stated otherwise: thresholds: VPm = POm = Rt = 0; excitatory synapses: G = 1 ms, G = 2 ms, tG = 0 (Golomb and Amitai 1997); GABAA synapses: A = 1 ms, A = 10 ms, tA = 3 ms; GABAB synapses: B = 40 ms, B = 150 ms, tB = 35 ms (Golomb et al. 1996; Kim et al. 1997); synaptic conductances: gVPm;Rt,A = 0.5, gVPm;Rt,B = 2, gPOm;Rt,A = 0.16, gPOm;Rt,B = 3.5, gRt;VPm = 1.2, gRt;POm = 1; and adaptation: ka,POm = 0.33 ms–1, ka,VPm = 0.1 ms–1, b,VPm = 10 ms; b,POm = 30 ms, kb,POm = kb,VPm = 0.05 ms–1, and a,VPm = a,POm = 100 ms.
Derived quantities
SPIKE NUMBER.
The function MPOm(t) (Eq. 3) is proportional to the instantaneous spiking rate of POm neurons. A measure that is of further utility in our analysis is the total number of spikes that are fired during a time interval by these neurons. This number is proportional to NPOmspikes, where
 | (14) |
 | (15) |
LATENCY.
The onset latency, t0, is computed in our analytical treatment. We further define the time between the start of the stimulus and the time that the instantaneous spiking rate, Mi(t), reaches half of its maximal value at the midpoint latency, as t0.5. This second definition is consistent with that used to quantify the latency in the experimental observations (Ahissar et al. 2000, 2001; Sosnik et al. 2001).
Computation
The delay differential equations and the auxiliary equations that define the model (Eqs. 1–12) are solved numerically by the Euler method with a time step of 
t = 0.02 ms.
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RESULTS |
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GABAB-mediated inhibition has slow kinetics and facilitates with increased presynaptic spiking. Under steady-state conditions, the slow kinetics cause a rhythmic input to lead to a delayed inhibition of thalamic cells. For a rhythmic stimulus, the extent of the delay will depend on the rate of rise of the brain stem input and the frequency of the stimulus. If the rise is gradual, facilitation will transform an increasing rate of stimulus-induced spiking into an increased latency in the response of thalamic cells because higher spike-rates lead to stronger inhibition. The model enables us to assess the consequences and stability of this mechanism and the specific role of the three thalamic nuclei.
Our first goal is to gain qualitative insight into the role of GABAB-mediated inhibition and the stimulus shape in the production of a frequency-dependent latency. To achieve this goal, we simplify the rate model to obtained a reduced, analytically solvable model. The analysis is extended to circuits with increasing level of complexity. Finally, the outcome of the reduced model is compared with simulations of the full rate model to demonstrate that the approximations used for the reduction do not disrupt the basic dynamical mechanisms. The simulations further allow us to make a detailed comparison between experimental and model results as well as to make predictions for the outcome of future experimental investigations.
Reduced model of the thalamic circuit
A simplified model is obtained through the inclusion of four assumptions. First, the GABAB-mediated decay-time B is larger than all the other synaptic time constants in the system. Therefore with the exception of the GABAB-mediated delay (which plays a special role in the dynamics) and decay, all of the synaptic processes are considered to be instantaneous. This allows us to focus on the special role of GABAB-mediated currents in shaping the thalamic dynamics. In particular, we set tPOm,delay = 0 in Eq. 13 because this time constant is much smaller than the GABAB decay time so that IPOm(t) = 
IVPm(t). Further, since the time constants of the fast excitatory synapses (G, G, and tG) are set to zero (Eqs. 5 and 6), we replace uVPm by MVPm and replace uPOm by MPOm in the expressions for the firing rate (Eqs. 2–4).
The second assumption concerns adaptation. Because adaptation does not affect the thalamic response at the onset of activity, we simply ignore it and take aPOm = aVPm = 0.
The third set of assumptions concern the fast conductances. GABAA-mediated inhibition is ignored because this inhibition has fast decay rate, which is now taken to be instantaneous. Therefore inhibition generated in response to a given stimulus decays before the next stimulus arrives and does not participate in generating the latency. The final assumption is that, for simplicity, the thresholds i are set to 0 for all nuclei i.
Based on the first assumption, the GABAB-mediated conductance in the Rt nucleus is simplified by taking the onset to be instantaneous, i.e., B = 0 (Eq. 9). This allows us to substitute xRt(t) by MRt,B(t – tB) (Eqs. 8 and 9). This yields
 | (16) |
To realize analytical treatment, we treat only cases where Tstim = 1/fstim >2tB. Furthermore, because the rise time of the brain stem stimulus-evoked response is 50 ms (Fig. 1A), which is only slightly greater than the delay of GABAB-mediated synapses, tB, we take tB to equal the stimulus duration. Further, to demonstrate the importance of this stimulus shape for the latency in a manner that avoids the algebraic complexity associated with a "double-ramp" treatment, we consider first a triangular stimulus to mimic the slowly rising phase and next a square stimulus to mimics the fast rising phase. The shapes of stimuli are, for 0 
t < Tstim
 | (17) |
 | (18) |
The value NPOmspikes, proportional to the number of spikes fired by POm neurons, is (Eq. 14)
 | (19) |
Analytical results for reduced POm-Rt circuits
POm-Rt CIRCUIT FOR TRIANGULAR STIMULUS: ESSENTIAL FEATURES FOR CODING STIMULUS FREQUENCY BY RESPONSE LATENCY.
We consider first a minimal circuit with feedback between the POm and Rt nuclei, in addition to input from the brain stem to POm thalamus (Fig. 1B). The dynamics of the GABAB activation variable, uRt,B(t), are determined by a single delay-differential equation with five parameters (Eq. 16) and the expression for the triangular stimulus (Eq. 17). Three of these parameters are time constants, i.e., the time period of the stimulus Tstim, the stimulus duration tB, and the GABAB decay rate B. The other two parameters are the POm-to-Rt excitatory conductance gRt;POm and the Rt-to-POm GABAB-mediated inhibitory conductance gPOm;Rt,B. The dynamics of activity in POm thalamus are determined by the difference between the excitatory stimulus IPOm(t) and the GABAB-mediated inhibition gPOm;Rt,B uRt,B(t) (Fig. 2A). Thus the latency for activation of POm neurons corresponds to the time when the difference between the stimulus and the inhibition becomes positive, namely (APPENDIX B, Eq. B2)
 | (20) |
 | (21) |
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 | (22) |
 | (23) |
). We examine the fixed point and its stability graphically (Fig. 2C). For small values of uRt,B(0), the corresponding value of uRt,B(Tstim) decreases with increasing values of uRt,B(0) as a result of heightened GABAB-mediated inhibition and leads to a concomitant decrease in the response of POm neurons to the stimulus. For the case of large values of uRt,B(0), the inhibitory feedback dominates and activity of POm thalamus is suppressed. The fixed point occurs when activity in POm thalamus is not fully suppressed. This point is stable below a critical value of GABAB-mediated conductance, e.g., gPOm;Rt,B 3.6 for the parameters of Fig. 2C. Last, although the reduced model has multiple parameters, the value and stability of the fixed point depends only on the products gRt,POm2gPom,Rt,B, gPOm;Rt,BuRt,B(0), and gRt,POm2/uRt,B(0). Thus an increase in gRt;POm can be offset by a decrease in gPOm;Rt,B and a rescaling of the value of the activity at the fixed point. The latency t0 is a monotonic function of gPOm;Rt,B up to the point of period doubling (Fig. 3A). The number of spikes per cycle NPOmspikes thus decreases with increasing stimulus frequency (Eq. 21; Fig. 2B). For values of gPOm;Rt,B that permit period doubling, i.e., 3.6 < gPOm;Rt,B < 7.1 for the parameters of Figs. 2C and 3A, the latency alternates with a period of two stimulus cycles. More complex behavior emerges for still larger values of gPOm;Rt,B. In general, we can evaluate the stability of the solution with a single stimulus cycle for arbitrary values of stimulus frequency, tB, and gPOm;Rt,B (Fig. 3B). This solution is stable for all frequencies for small gPOm;Rt,B and is unstable for stimulus frequencies above, roughly, fstim = 2 Hz for large values of gPOm;Rt,B. The dependence of the latency t0 on the stimulus frequency fstim and gPOm;Rt,B is shown in Fig. 3C. For values of gPOm;Rt,B that do not lead to period doubling, the latency increases with fstim and can be even somewhat larger than tB/2 but not close to tB. Thus the minimal model captures the essential scaling of increased latency with increased stimulation frequency, albeit for a constrained set of parameter values.
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) (APPENDIX C). This behavior lies in the coupled activation of the Rt and POm nuclei. We consider how excitation of Rt by VPm thalamus, rather than solely through POm thalamus, can decouple the activity of the Rt and POm nuclei, lead to more robust dynamics, and enable large latencies up to tB. We begin with an analytically tractable model that incorporates two additional features over the POm-Rt circuit: solely feedforward excitatory connections from VPm to RT thalamus, with conductance gRt;VPm (Fig. 1C), and brain stem input the shape of which is identical for the two nuclei but the amplitude of which for POm thalamus is taken to be weaker than that for VPm thalamus, that is
 | (24) |
To gain insight into the role of VPm input to Rt thalamus, and to examine latency as a function of the coupling between these nuclei gRt;VPm, we first consider the circuit with POm-to-Rt feedback excitation turned off, i.e., gRt;POm = 0. For this case, the steady state with the period of the stimulus is always stable, but the output of POm thalamus is silent above the value gRt;VPm*. For both triangular (Eq. 17) and rectangular stimuli (Eq. 18), the spiking activity decays gradually from a maximal value to zero; this is shown for fstim = 8 Hz in the examples of Fig. 5, A and B. For the triangular stimulus, the latency increases gradually from 0 to tB (Fig. 5A). In contrast, for the rectangular stimulus, the latency remains zero for an extended interval of values of gRt;VPm that satisfy gPOm;Rt,B u(0) 1; note that u(0) increases gradually with gRt;VPm. The latency then jumps steeply and reaches t0 = 1 at gRt;VPm*, for which gPOm;Rt,B u(0) = exp(tB/B) (Fig. 5B). Therefore for a rectangular stimulus, the latency increases with frequency only for a restricted range of values of gRt;VPm for which POm thalamic activity is very small. This implies that a gradual increase of the stimulus activity with time is necessary to code stimulus frequency by latency for this purely feedforward architecture.
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-1gPOm;Rt,B for VPm versus POm thalamus, respectively. The former value should be the smaller of the two to ensure that the latency in VPm thalamus is essentially constant. Numerical results for the full POm-Rt-VPm circuit
The analytical theory suggests an explanation for the latency coding. Still, several questions still remain open. First, are the analytical results valid despite the approximations that were made? Second, what is the effect of the dual-sloped rising phase of the brain stem stimulus, as observed experimentally (Fig. 1A), as opposed to just a triangular shape? Third, can we explain the detailed shape of the PSTHs recorded in the POm and the VPm nuclei? Specifically, why does the POm activity decay rapidly with time after the initial onset at low stimulus frequency, whereas such decay is seen neither in the POm in response to high-frequency stimulation nor in the VPm?
To answer the preceding questions and to draw a close comparison between responses calculated for the model and those observed in experiments, we have carried out numerical simulations of the full rate model (APPENDIX A) with the architecture of Fig. 1D. We incorporate inhibitory feedback that satisfies gPOm;Rt,B > gVPm;Rt,B; adaptation in the activity of POm thalamus, and weaker adaptation in VPm thalamus, to account for the reduction in the population of neurons that are available to spike after each stimulus (Eggert and van Hemmen 2001 2000); and a dual-sloped rising phase for the stimulus (Fig. 7, A and B).
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GABAB-mediated inhibition by Rt thalamus becomes strongly activated with increased values of the stimulation frequency above fstim = 2 Hz (Fig. 7C). The excitatory stimulus minus the GABAB-mediated inhibition is positive only near the peak of the stimulus, where the stimulus rises gradually, as in the case of a triangular stimulus. As a result, the activity in POm thalamus is strongly suppressed just after the onset of the stimulus (Fig. 7E). With increasing time, the brain stem input overcomes the GABAB-mediated inhibition and the activity in POm rises until the stimulus ends. For times 50–90 ms after the stimulus onset, the GABAB synaptic activation is balanced by the decreasing adaptation and the activity in POm thalamus is essentially independent of fstim (Fig. 7E). These dynamics result in a calculated instantaneous spike rate whose latency to peak and overall amplitude compares favorably with those in the experimental data (cf. Figs. 7E with 1A).
With regard to VPm thalamus, the slow GABAB-mediated inhibition reduces the overall activity with increasing stimulus frequency (Fig. 7F). Yet, as a result of a larger input from the brain stem ( = 0.6 in Eq. 13) and the weaker GABAB-mediated inhibition compared with the case for POm thalamus, the excitatory stimulus minus the GABAB-mediated inhibition is positive everywhere except just after the onset of the stimulus. The reduction of activity by GABAB-mediated inhibition decreases the firing activity just following the stimulus onset but is insufficiently strong to affect the latency of the response. Thus there is negligible change in the latency to peak, with increasing stimulus frequency. As in the case of POm thalamus, the time dependence of the instantaneous spike rates in our simulations compares favorably with those in the data (cf. Figs. 7F with 1A).
COMPARISON OF THEORY WITH EXPERIMENT: LATENCY AND SPIKE COUNT. The calculated and observed latencies were expressed in terms of t1/2, which is the time for the instantaneous rates of spiking, MPOm(t) and MVPm(t), to reach one-half of their maximum value. The values of t1/2 computed from the model well approximate the observations for both the POm and VPm nuclei (Fig. 7, G and H). Further, the calculated increase in the value of t1/2 with increasing stimulus frequency for POm thalamus is a robust, monotonic function of gPOm;Rt,B up to the frequency where POm neurons are silenced by inhibition (Fig. 7I). In contrast to the close match of theory and observation for the latency effect, the calculated values of the number of spikes per cycle in POm thalamus decreases more sharply with frequency than in the observations (Figs. 7G). This deviation results from adaptation at low frequencies of stimulation that is relatively weak in the model. Nonetheless, the overall match between calculated and observed values (Fig. 7, E–H) lends support to the idea that the heightened frequency-dependent latency and reduced spike-count in POm versus VPm thalamus results from a stronger GABAB-mediated inhibitory feedback, a weaker brain stem input for neurons in POm thalamus (Eq. 13), and the double-ramp, monotonically increasing brain stem stimulus.
EFFECT OF STIMULUS DURATION AND SHAPE. We seek to account for the effect of stimulus width on the latency effect in POm thalamus and further predict the nature of the spiking response in the POm and VPm nuclei for arbitrary stimuli. Intuitively, the strongly facilitating nature of the GABAB-mediated currents should make the rate of spiking sensitive to the duration of the stimulus.
We first consider variations in the total duration of the stimulus (Fig. 8A). The period of the initial rise was kept fixed but the duration of the second rising phase and the falling phase were co-varied. The duration was parameterized by a multiplicative factor, denoted 
, where = 1 is the value for a stimulus with a total rising phase of 50 ms (Fig. 8A1). The change in time course of MPOm(t) and MVPm(t) as a function of frequency is relatively weak for the case of a brief stimulus, i.e., = 0.4 for a rise-time of 23 ms (Fig. 8A, 2 and 3), compared with a long stimulus (Fig. 7, E and F). The calculated time course compares favorably with those recorded with a 20 ms stimulus (Fig. 7 in Sosnik et al. 2001). In general, the range of the frequency-dependent latency for spiking in POm thalamus is predicted to increase as the duration of the stimulus increases toward its full width for = 1 (Fig. 8A4).
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; Sosnik et al. 2001) as the former method typically involves a much more rapid initial movement. |
DISCUSSION |
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Comparison with experimental results
Analysis of our model reveals that the latency increases with frequency in POm thalamus but not in the VPm thalamus if the brain stem input to the VPm is larger than the brain stem input to the POm and the GABAB–mediated inhibition is stronger in the POm than in the VPm. The first condition is clearly seen in in vivo experiments because the firing rate of VPm neurons in low stimulus frequencies is considerably higher in the VPm than in the POm (Diamond 1995; Diamond et al. 1992; Sosnik et al. 2001). Furthermore, in recent in vitro experiments, C. E. Landisman and B. W. Connors (unpublished data) found that VPm neurons exhibit higher maximal firing rates from POm neurons when stimulated with injected current. These researchers also found that VPm cells were 2.5 times less likely than POm cells to have a GABAB receptor-like component as part of their inhibitory responses. Hence these recent experimental results support our view of the role of GABAB–mediated inhibition in coding frequency by latency in the POm but not in the VPm.
We find that strong feedback connections between POm and Rt thalamus destabilize the steady-state activity with the period of the stimulus via a period doubling bifurcation. Analysis of the experimental data (Sosnik et al. 2001) (APPENDIX C) did not reveal any indication for activity with double period. Recent observations (Webber and Stanley 2004) have revealed period doubling in cortical activity in response to stimuli with duty cycles of 0.5 and frequencies between 4 and 8 Hz. This behavior may reflect similar behavior in the thalamic input to cortex. The stimuli used by those authors are more prolonged than the stimuli analyzed here and may generate a strong GABAB-mediated inhibitory response.
The GABAB kinetics used in the model (Eqs. 9 and 10) are based on the results from dual recordings (Kim et al. 1997). These observations show that a single presynaptic neuron should fire a high-frequency burst of action potentials to evoke a substantial GABAB-mediated response from the postsynaptic neuron. Indeed, Rt cells fire at a high rate in comparison with thalamic relay nuclei (Hartings et al. 2000). Furthermore because GABAB–mediated synapses are metabotropic and involve the cooperative activation of G proteins (Destexhe and Sejnowski 1995; Golomb et al. 1996), near-synchronous spiking of several presynaptic cells may act cooperatively to facilitate the stronger GABAB-mediated response.
Consequences of approximations in our model
Single-cell properties may well affect the latency and adaptation effects in the POm nucleus. Although there is little data on the cellular properties of neurons in POm thalamus, the lateral posterior (LP) nucleus is an analogous nucleus in the visual stream (Peters 1985) and the nonlemniscal nuclei are analogous nuclei in the auditory stream (Yu et al. 2004). In vitro intracellular recordings from neurons in the LP nucleus revealed significant potassium A-type and calcium T-type currents (Li et al. 2003). The A current can further delay the occurrence of the first spike in a stimulus cycle, whereas the combination of a T current and inhibitory input may contribute to the production of a burst of spikes followed by an afterhyperpolarization that terminates neuronal activity (Sherman 2001; Steriade et al. 1993). These cellular properties are likely to affect the details of our analysis of the full model (Fig. 7), but not our conclusions regarding the latency effect (Figs. 2–6).
The difference in latency coding between the VPm and POm thalamus is also reflected by neurons in cortical layers that receive axonal projection from these nuclei, i.e., layers IV and Va, respectively (Ahissar et al. 2000, 2001; Ahrens et al. 2002). Cortical feedback will act to supplement the brain stem input to thalamic relay cells (Deschenes et al. 1998; Diamond et al. 1992), and especially to excite Rt thalamus (Golshani et al. 2001). Recalling that thalamocortical (Gil and Amitai 1996) and even corticothalamic (Swadlow 1994) synaptic connections are much faster than the decay time of GABAB-mediated synapses, we anticipate that the introduction of cortical effects into the present thalamic model can, to a first approximation, be accomplished by a variation of the strength of the synaptic connections within the present architecture (Fig. 1D). As a test of this conjecture, we examined a circuit that included cortical feedback such that the Rt thalamus received all its input from excitatory cortical cells (APPENDIX E). We found that the dynamics of this circuit was similar to that of the solely thalamic circuit (Fig. 7) in support of our reductionism. Other factors, such as GABAB-mediated synapses within the neocortex (Garabedian et al. 2003), may influence the latency effect in POm thalamus at high frequencies of stimulation.
The POm nucleus is innervated by GABAergic neurons from zona incerta (ZI) (Bartho et al. 2002; Lavallee et al. 2005; Power et al. 1999; Trageser and Keller 2004) and from the anterior pretectal (APT) nucleus (Bokor et al. 2005). The ZI and the APT nuclei are innervated by trigeminal inputs, and the ZI is also innervated by layer 5 cortical projections. It is not clear whether ZI or APT can play the role of Rt thalamus in shaping the latency since there is currently no experimental evidence for a GABAB-mediated connection from these nuclei to thalamic nuclei. From the perspective of temporal dynamics, the GABAA-mediated inhibition from neurons in ZI and in APT nucleus (Bokor et al. 2005) plays the same role as the GABAA-mediated inhibition from Rt neurons, because the brief delay or the brief advance of inhibition is negligible in comparison with the relevant time scale of order of tens of milliseconds.
The analytically solvable model was developed from the full model (Eqs. 1–13) using several approximations: the GABAB-mediated decay-time, B, is larger than all the other synaptic time constants in the system; adaptation is ignored; GABAA-mediated inhibition is ignored; and the thresholds, i, are set to 0. We numerically simulated the full model to show that the results of an analytically solvable simplified model did not depend on the preceding approximations. Specifically, the circuit produces an increase in latency with increasing frequency when GABAA-mediated synapses and adaptation are included. The simulations show that GABAA-mediated inhibition and adaptation bring the temporal structure of the solution of the model closer to the observed structure.
Adaptation and depression
Adaptation is incorporated in the full model to account for the fast decay of spiking activity in POm but not VPm thalamus at low stimulus frequencies. The adaptation process in the model (Eqs. 2, 3, 11, and 12) is multiplicative. Qualitatively, this process takes into account the fact that a cell that has just fired cannot fire anymore until the adaptation effects vanish. Biophysically, this process may be based on the activation of the T-type Ca2+ current that leads to a burst of one or more spikes followed by a prolonged refractory period. Interestingly, POm neurons are more prone to burst mode activity than VPm neurons (Ramcharan et al. 2004), consistent with the stronger adaptation they exhibit in vivo. Adaptation may also result from the activation of slow K+ currents. In previous rate models (Hansel and Sompolinsky 1998; Tabak et al. 2000), adaptation was modeled as a subtractive process. However, we could not mimic the strong adaptation at low fstim with such a process.
In principle, the role of GABAB-mediated synapses proposed here can be replaced by synaptic depression of the brain-stem-to-POm excitatory synaptic connections or by intrinsic adaptation with the proper time scale, mediated, for example, by slow Ca2+-activated K+ currents or other long-lasting cellular adaptation processes. We are not aware of any experiment in which the dynamic properties of the brain-stem-to-POm synapses were quantified. A depression-based mechanism will likely require that the depression of the brain-stem-to-POm synapses be considerably stronger than the depression of the brain-stem-to-VPm synapses. A mechanism based on intrinsic adaptation may require a second adaptation process in addition to the one that is responsible for the gradual decline of POm activity at low frequency. Because mechanisms based on depression or adaptation do not include an interaction between the POm and VPm nuclei, it is expected that the steady state with the period of the stimulus will undergo a period doubling bifurcation if the latency at high frequencies is large enough.
Predictions from the model
Our analysis yields several predictions for future experiments. First, the complete blockade of GABAB-mediated synaptic transmission in POm thalamus will eliminate the increase of latency with increasing stimulus frequency (Fig. 7I). Second, barring adaptation effects, the blockade of GABAB-mediated synaptic transmission in either the POm or VPm nuclei should eliminate the decrease in spike rate with increasing frequency. In partial support of this claim, Castro-Alamancos (2002a) blocked both GABAA- and GABAB-mediated inhibition in VPm thalamus and found that the spiking activity no longer decreases with the frequency of stimulation. Third, partial blockage of GABAB-mediated inhibition will preferentially eliminate the increase in latency but not the decrease in spiking with increasing frequency. Fourth, we argued that activation of Rt thalamus via excitatory input from VPm thalamus, which acts as feedforward inhibition from the VPm to the POm nucleus, is essential for a robust latency effect. Thus blockage of activity in VPm thalamus should suppress the latency effect in POm thalamus (Fig. 6). Last, our analysis emphasized the importance of the shape of the spiking pattern in brain stem, which follows the time course of vibrissa movement (Deschenes et al. 2003; Jones et al. 2004; Lichtenstein et al. 1990), in the latency effect. We expect that the latency effect will be absent if deflection of the vibrissae occurs too quickly (Figs. 5B and 8B) or is too brief (Fig. 8A).
Comparison with a phase-locked loop algorithm
The dependence of the latency in POm thalamus on the stimulus frequency has been shown to be qualitatively consistent with the behavior of a phase-locked loop (PLL) algorithm (Ahissar and Kleinfeld 2003; Ahissar et al. 2000). The implementation of the algorithm involves negative feedback to neurons in POm thalamus from neuronal oscillators in cortex or thalamus. In principle, a thalamocortical PLL circuit is similar to our reduced POm-Rt circuit (Fig. 1B) but with an oscillator replacing Rt. In both models, the latency of the response in POm thalamus is determined by the interaction between the inhibitory feedback signal and the brain stem input. However, the dependency of the latency on the frequency stems from different sources. In the PLL model, it arises from a phase correction of the oscillator in Rt thalamus, whereas in the present neuronal model, it arises from a gradual increase in the amplitude of the GABAB-mediated inhibition. Thus for example, a rectangular brain stem stimulus is expected to yield latency coding within a PLL circuit but not with a GABAB-mediated mechanism (Fig. 5B).
Computational relevance of Rt thalamus
Experimental (Steriade et al. 1993; Von Krosigk et al. 1993) and theoretical (Golomb et al. 1996) studies have demonstrated the importance of the Rt nucleus in the generation of brain oscillations, such as spindles. The Rt nucleus is also a locus for epileptic seizures (McCormick and Contreras 2001). Here we suggest a computational role for the Rt nucleus in somatosensory information processing: it employs GABAB-mediated inhibition to code stimulus frequency by the latency of the spike response in POm thalamus. The Rt nucleus can modulate sensory inputs using GABAB-mediated synapses if the stimulus is strongly temporally-modulated in the time scale of the decay of GABAB-mediated inhibition, namely the stimulus frequency should be 5–10 Hz. A more complex role that involves GABAB-mediated inhibition, such as the detection of a change in timing or amplitude in an otherwise periodic sensory stream (Mehta and Kleinfeld 2004), remains to be determined.
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APPENDIX A: PARAMETERS OF BRAIN STEM INPUT |
|---|
 | (A1) |
For t > Tstim, the brain stem input is IVPm(t) = IVPm(t – kTstim), where k is the largest integer smaller than t/Tstim. The brain stem input to POm thalamus is proportional to the brain stem input to the VPm, and it is delayed to allow larger latency in POm even for low stimulus frequencies
|
| (A2) |
= 0.6, tPOm,delay = 7 ms. The values used for Fig. 7 are: q0 = 0, q1 = 6 ms, I0 = I1 = 0, q2 = 11 ms, I2 = 0.8, q3 = 56 ms, I3 = 1.5, q4 = 96 ms, I4 = 0. The starting times of the first ramp for the inputs to the VPm and POm nuclei, q1 and q1+ tPOm,delay, were chosen such that the latencies t0.5 in the two nuclei, obtained from the model at low stimulus frequencies, fit the latencies observed experimentally.
The following parameters are used for Fig. 8.
A) The width of the second rising phase and the decaying phase together is times the reference values of these phases, respectively. The parameters are q0 = 0, q1 = 6 ms, q2 = 11 ms, I0 = I1 = 0, I2 = 0.8, q3 = q2 + x 45 ms, I3 = I2 + x 0.7, q4 = q3 + x 40 ms, I4 = 0.
B) The slope of the first rising phase is dI/dt = 0.16 ms–1; I2 = (dI/dt) (q2 – q1). The other parameters are: q0 = 0, q1 = 6 ms, I0 = I1 = 0, q3 = 56 ms, I3 = 1.5, q4 = 96 ms, I4 = 0. The values of q2 are 15.4 ms for Fig. 8, B1 (solid line), B2, B3; 11 ms for B1 (dotted line); and 19.8 ms for B1 (dashed line).
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APPENDIX B: ANALYSIS OF THE REDUCED POM-RT MODEL |
|---|
 | (B1) |
Our goal is to find uRt,B(Tstim), the GABAB-mediated synaptic activation after one period, knowing uRt,B(0). The variable uRt,B is affected by the stimulus at time t if IPOm(t – tB) – gPOm;Rt,B uRt,B(t – tB) > 0 (Eq. B1). We denote the onset time of activity of POm neurons by t0. From now on, the analysis depends on the stimulus shape.
Triangular stimulus (Eq. 17)
The POm onset time t0 is determined by the equation (Fig. 2A)
 | (B2) |
This equation is valid only if t0 < tB, namely uRt,B (0) < uRt,B;max = 
. If uRt,B(0) > uRt,B;max, uRt,B decays exponentially during the whole period
 | (B3) |
To calculate uRt,B(Tstim) for smaller uRt,B(0), we note that uRt,B(tB + t0) = 
, and integrate Eq. B1 to obtain uRt,B(2tB)
 | (B4) |
 | (B5) |
 | (B6) |
Equations B3 and B6 define a map from the value of the GABAB-mediated conductance variable uRt,B(0) at the beginning of one stimulus to the value of uRt,B(Tstim) at the beginning of the next stimulus. The intersection between the map (Eqs. B3 and B6) and the diagonal line uRt,B(Tstim) = uRt,B(0) is the fixed point of the map, denoted by 
. This fixed point corresponds to uRt,B being periodic with a period of a single stimulus cycle. The fixed point is stable if (Berge et al. 1984)
 | (B7) |
The fixed point undergoes a period doubling (PD) bifurcation if the derivative in Eq. B7 equals –1. The value uRt,B(Tstim) depends on uRt,B(0) explicitly and also implicitly through the latency t0 (Eq. B2). Therefore
 | (B8) |
By differentiating Eq. B4 with respect to t0 and using Eqs. B5 and B2, one sees that 
uRt,B/t0 = 0. Therefore duRt,B(Tstim)/duRt,B(0) = uRt,B(Tstim)/uRt,B(0), and the PD bifurcation point is given by
 | (B9) |
).
The quantity NPOmspikes, proportional to the number of spikes fired by POm neurons (Eq. 19, Fig. 2A) can be calculated exactly. We find
 | (B10) |
Rectangular stimulus (Eq. 18)
According to Eqs. B1 and 18, three behavioral regimes are determined by the initial value gPOm;Rt,B uRt,B(0).
First, if gPOm;Rt,B uRt,B(0) 1, the POm onset time is t0 = 0, and
 | (B11) |
Second, if 1 < gPOm;Rt,B uRt,B(0) < 
, the POm onset time is t0 = B log[gPOm;Rt,B uRt,B(0)], and
 | (B12) |
Third, if gPOm;Rt,B uRt,B(0) 
, uRt,B decays exponentially during the whole period according to Eq. B3.
Numerical simulations show that the fixed point of the map uRt,B(Tstim) versus uRt,B(0) is given by the intersection of the line uRt,B(Tstim) = uRt,B(0) with Eq. B11. The PD bifurcation point is given by
 | (B13) |
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APPENDIX C: TESTING FOR A PERIOD-2 BEHAVIOR IN THE DATA |
|---|
). For comparison, we also re-examined 23 multiunit recordings from the VPm thalamus (Sosnik et al. 2001). If the multiunit recording exhibits period-2 behavior, the average absolute value of the difference in the number of spikes between the response to a first stimulus and the response to the subsequent stimulus should be large in comparison to the absolute value of the difference in the number of spikes between the response to a first stimulus and the response to the second subsequent stimulus. To quantify this behavior, we denote the number of spikes fired by the population in response to the ith stimulus by ri. In practice, each train of stimuli lasts for 3 s, and we drop the response for the first second to avoid transient effects. In the remaining 2 s, there are NT stimuli. For example, if fstim = 8 Hz, NT = 16. We define the value Si as (Fig. C1A)
 | (C1) |
|
 | (C2) |
The trial-average (over 24 trials) and the SD of Y for a specific multiunit recording and a specific period are denoted by 
Y
and 
Y, respectively. This analysis is an alternative to spectral analysis.
The Z score of the distribution is defined as
 | (C3) |
The value of Z should be large compared with 1 if the unit exhibits doublet behavior. Figure C1 B–G,, shows the distributions P(Z), computed for the POm and VPm populations of units and for fstim = 5, 8, and 11 Hz. The distributions P(Z) for both POm and VPm units for all those frequencies are centered around zero. All the values of Z were <1, except for one case of a POm unit and fstim = 8 Hz, where Z = 1.2. We conclude that there is no statistically significant indication for doublet behavior in the data.
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APPENDIX D: ANALYSIS OF THE VPm-POm-Rt CIRCUIT WITH FEEDFORWARD VPm-TO-Rt CONNECTIONS |
|---|
For the VPm-POm-Rt circuit without Rt-to-VPm feedback connections (Fig. 1C), Eq. 16 for the GABAB-mediated synaptic activation from the Rt becomes
 | (D1) |
For a triangular stimulus (Eq. 17), the POm term in Eq. D1 is nonzero for t0 < t – tB < tB, where t0 is given by Eq. B2. Therefore
 | (D2) |
 | (D3) |
 | (D4) |
The stability of the fixed point uRt,BFP = uRt,B(Tstim) = uRt,B(0) is given by Eqs. B7 and B8. Using Eqs. D2 and D3, one shows that uRt,B(Tstim)/t0 = 0. Therefore duRt,B(Tstim)/duRt,B(0) = uRt,B(Tstim)/uRt,B(0), and the PD bifurcation point is given by
 | (D5) |
When the value gRt;VPm reaches the value gRt;VPm*, t0 = tB, the POm thalamus is silent, and its contribution to Eq. D3 is zero. Therefore gRt,VPm* does not depend on gRt;POm, and is given by (Figs. 5 and 6)
 | (D6) |
Rectangular stimulus (Eq. 18).
For simplicity, we limit ourselves to a purely feedforward VPm-POm-Rt circuit: gVPm,Rt;B = gRt,POm = 0 (Fig. 1C), Using Eqs. D1 and 18, we obtain the fixed point value
 | (D7) |
For gPOm,Rt,BuRt,BFP 1, t0 = 0 and
 | (D8) |
 | (D9) |
|
APPENDIX E: CORTICAL FEEDBACK |
|---|
 | (E1) |
 | (E2) |
 | (E3) |
|
CxE = 0, CxI = 0. The dependencies of the waveforms of the GABAA and GABAB inhibition and the VPm and POm spiking activity on fstim (Fig. E1, B–E) are very similar to those we computed for the thalamic-only circuit (Fig. 7, C–F). Similarly, the dependencies of the latencies t1/2 and the number of spikes per cycle in the POm and the VPM on fstim are also very similar in the two circuits (Figs. E1, F and G, and 7, G and H). The reason behind this resemblance is that the dependence of the system dynamics on fstim is governed by the slow Rt-to-POm and Rt-to-VPm GABAB dynamics. These dynamics are dependent on the Rt activity during the cycle, and it does not matter whether the Rt is activated by the cortical feedback or directly by the thalamic relay nuclei.
|
GRANTS |
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|
NOTE IN PROOF |
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|
ACKNOWLEDGMENTS |
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|
FOOTNOTES |
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Address for reprint requests and othercorrespondence: D. Golomb, Dept. of Physiology, Faculty of Health Sciences, Ben-Gurion University of the Negev, PO Box 653, Beer-Sheva 84105, Israel (E-mail: golomb{at}bgu.ac.il)
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ABSTRACT
INTRODUCTION
) denotes the period doubling (PD) point; the solid lines (—) correspond to stable fixed points. B: stability of the fixed point with a period of a single stimulus cycle as a function of the GABAB conductance gPOm;Rt,B and the stimulus frequency fstim. The PD line separates the parameter regime in which the fixed point with the periodicity of the stimulus is stable (below —) from the regime in which it is unstable (above —). C: normalized latency t0/tB as a function of GABAB-mediated conductance gPOm;Rt,B and the stimulus frequency fstim. The latency was calculating by averaging over 50 cycles after initial 950 cycles. —, the PD line. The latency is zero in the regime where the fixed point with a period of a single stimulus cycle is stable.
and 