INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Considerable attention has been focused in recent years on the functioning of thalamic relays, because it has become clear that the thalamus does not serve as a simple, machine-like relay of information to cortex (for recent reviews, see Sherman and Guillery 1996, 1998). Instead the thalamus controls the extent and nature of information being relayed in a dynamic fashion that appears to be related to behavioral state and perhaps attentional demands. A good example is the lateral geniculate nucleus, the thalamic relay of retinal information to visual cortex. Only 5-10% of synapses on geniculate relay cells derive from retina. The rest derive from nonretinal sources, including local GABAergic cells, feedback afferents from visual cortex, and various pathways from the brain stem, and these modulate the nature of retinogeniculate transmission (Sherman and Guillery 1996, 1998).

In addition to the complexity of thalamic circuitry, the membrane properties of relay cells contribute to the nature of the relayed information. In particular, thalamic relay cells exhibit a voltage- and time-dependent, low-threshold, transient Ca²⁺ conductance, that, when activated, allows Ca²⁺ to enter the cell via T-type (for "transient") Ca²⁺ channels, producing a transmembrane current, I_T, and leading to a large depolarization known as the low-threshold Ca²⁺ spike. The inactivation state of I_T determines whether information is relayed to cortex in tonic mode or burst mode (Jahnsen and Llinas 1984a,b; Sherman 1996). When the cell starts off relatively depolarized (above roughly 60 mV for >50-100 ms), I_T is inactivated, and the relay cell responds to an excitatory input [e.g., a retinal excitatory postsynaptic potential (EPSP)] with sustained firing of unitary action potentials. This is the tonic firing mode. However, if the cell is hyperpolarized first (below roughly 65 mV for >50-100 ms), the inactivation of I_T is removed (i.e., I_T becomes deinactivated), and now a sufficient depolarization or EPSP will activate I_T. The result is a low-threshold Ca²⁺ spike with a brief burst of 2-10 action potentials riding its crest.

One of the keys to understanding how thalamic relays work is to understand in more detail how the input/output properties of relay cells are affected by the inactivation state of I_T. We sought to do this with both an experimental and modeling approach. By recording from relay cells of the lateral geniculate nucleus of the cat in vitro, we measured input/output properties by injecting into the cell sinusoidal currents that varied in amplitude, frequency, and mean level, and we performed analogous input/output experiments on a minimal relay cell model to test the degree to which the essential features of I_T accounted for relay cell responses. In addition to providing an easily parameterized set of stimuli that lends itself to Fourier analysis, the use of sinusoidal current injection allows us to interpret our results in the context of the spatial and temporal frequency analysis paradigm that has such a successful history in visual systems neuroscience (for review, see Shapley and Lennie 1985). Of course, our use of Fourier techniques is not based on an assumption of the linearity of relay neuron responses but simply reflects an historically preferred method of extracting relevant measures of cellular response (see METHODS).

For the theoretical component of this study, we developed a minimal "integrate-and-fire-or-burst" (IFB) neuron model. This model is constructed by adding a slow variable representing the deinactivation level of I_T to a classical integrate-and-fire neuron model (Knight 1972). The IFB model is designed specifically to be as simple as possible while still quantitatively reproducing much of the empirically observed properties of the relay cells. One motivation for developing such a minimal model is to simplify the parameter selection process. Furthermore, because the IFB model is minimal, a detailed characterization of its response properties leads to insight regarding the stimulus dependence of burst versus tonic response modes in thalamic relay cells. A final motivation for development of the IFB model is to have a realistically tuned yet computationally undemanding relay cell model that can be used in large scale network simulations of thalamic function.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Experimental methods

We performed intracellular recordings with the whole cell configuration on thalamic relay cells of young cats (5-8 wk of age) in compliance with approved animal protocols. We used a thalamic brain-slice preparation containing the lateral geniculate nucleus. Briefly, the animals were anesthetized deeply with 25 mg/kg ketamine and 1 mg/kg xylazine and a block of tissue containing the thalamic region was removed and placed in cold, oxygenated slicing solution containing (in mM) 2.5 KCl, 1.25 NaH₂PO₄, 10.0 MgCl₂, 0.5 CaCl₂, 26.0 NaHCO₃, 11.0 glucose, and 234.0 sucrose. Thalamic slices (250-300 µm) were cut in a coronal or sagittal plane with a vibrating tissue slicer and placed in a holding chamber (30°C) for >2 h before recording. Individual slices were transferred to a submersion-type recording chamber maintained at 30°C and continuously perfused with oxygenated physiological solution containing (in mM) 126.0 NaCl, 2.5 KCl, 1.25 NaH₂PO₄, 2.0 MgCl₂, 2.0 CaCl₂, 26.0 NaHCO₃, and 10.0 glucose, all at pH 7.4.

We used an Axoclamp 2A amplifier to obtain current-clamp recordings from geniculate relay neurons in the A-laminae, and we continuously monitored the bridge balance throughout the recordings. The recording pipette solution contained (in mM) 117.0 K-gluconate, 13.0 KCl, 1.0 MgCl₂, 0.07 CaCl₂, 0.1 EGTA, 10.0 HEPES, and 0.5% biocytin. Data were digitized, stored on-line using Axotape software (Axon Instruments), and also recorded onto VHS tape for off-line analysis. Current injection through the recording electrode consisted of a sinusoidal waveform with an AC component (I₁) that varied in both amplitude (50-800 pA) and frequency (0.1-100 Hz). The DC component (I₀) of the current waveform was altered to manipulate the firing mode of the neuron (i.e., burst vs. tonic). All experimental records of membrane potential of relay neuron in whole cell mode have been adjusted to account for a 10-mV junction potential.

Fourier analysis of experimental and theoretical responses

Customized user M-files were written for MATLAB 5.2.0 (The MathWorks) to perform data analysis using an SGI Challenge supercomputer that runs the IRIX operating system. For each stimulus condition, a periodic histogram (q_k, k an integer, 0 < k < N 1, n = 64 bins) was constructed that tallied over c cycles of period T the number of action potentials (q_k) evoked by the experimental or model relay neuron at each of N blocks of phase relative to the applied current's period. Accounting for the number of cycles recorded and the time represented by one bin of phase (i.e., for 64 bins, 1 bin is /32 rad), we generated the (periodic) poststimulus response histogram (PSTH) defined by Q_k  q_kN/cT. A discrete Fourier transform of this PSTH was performed, leading to a set of N complex valued numbers, _n, given by (Press et al. 1992)

(1)

each with an associated amplitude, A_n = |_n|, and phase, P_n = arg(_n)/2. With the preceding definitions, F₀ = A₀/N has units of spikes/second and is the mean firing rate of the neuron; that is, F₀ = q_tot/cT, where q_{tot  k} q_k is the total number of spikes during the trial. The fundamental or stimulus-driven component of the response, F₁, is given by F₁ = (A₁ + A_N1)/n = 2A₁/N. (For the 2nd equality, we have used A₁ = A_N1, which follows because the raw histogram, q_k, is real valued). As defined in the preceding text, P₁ is the phase advance or lag of this stimulus-driven response, has units of cycles, and takes values between 0.5 and +0.5. In addition, we define the following "index of nonlinearity"

(2)

This index of nonlinearity, , is the normalized power of all modulated (non-DC) components of the PSTH not accounted for by the fundamental component of the response. Thus  takes values between zero (completely linear) and one (completely nonlinear). F₀, F₁, P₁, and  are the response measures of relay cells on which we shall concentrate.

For clarity of figure presentation, we also normalized the raw histogram, q_k, by the total number of spikes during the trial (q_tot). The result is a spike phase density histogram (SPDH), defined by _k  q_k/q_tot. This SPDH has unit area and represents the likelihood of observing an action potential at a particular phase of the applied current.

IFB model

GENERAL FEATURES. The IFB model is constructed by adding a slow variable to a classical integrate-and-fire model neuron. The slow variable, h, represents the inactivation of the low-threshold Ca²⁺ conductance, which involves T-type Ca²⁺ channels and produces a transmembrane current, I_T. The model equations are (Rinzel 1980)

(3)

(4)

The current balance equation, Eq. 3, includes the sinusoidal applied current, I_app =I₀ + I₁ cos(2f t); a constant conductance leakage current (I_L) of the form, I_L = g_L(V  V_L); and the low-threshold Ca²⁺ current, I_T. An action potential occurs whenever the membrane potential reaches the suitable firing threshold (V) such that V(t) = V  V(t⁺) = V_reset.

(5)

PARAMETER SELECTION. Standard parameters were selected for the IFB model in the following fashion. First, experimental observations indicated that the resting membrane potential of the relay neurons recorded in vitro was 75 to 65 mV. In the absence of applied current (I_app), the leakage term (I_L) exclusively sets the resting potential of the neuron model. We thus set V_L to 65 mV. A second experimental observation is that the relay neurons recorded in vitro are hyperpolarized sufficiently at rest so that I_T is deinactivated. Indeed, quiescent relay neurons in the slice responded to a brief depolarization with a burst of action potentials. For this reason, we set V_h to 60 mV, ensuring that the threshold for activation (and deinactivation) of I_T is several millivolts greater than V_L, as observed experimentally. This value for V_h also roughly corresponds to the observed threshold for activation of bursts.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Current-clamp recordings from a total of 12 relay neurons from the lateral geniculate nucleus of the cat were included in the present study. All cells displayed physiological characteristics consistent with healthy relay neurons, including a hyperpolarization-activated sag current (I_h), burst discharge, and overshooting action potentials. Cells included in this study had an average resting membrane resting potential of 67.8 ± 4.1 mV (mean ± SD) and an input resistance that averaged 153.0 ± 38.0 M.

The preferred firing mode of the relay celltonic or burstwas controlled largely by constant current injection from the recording pipette. This constant current injection is referred to in the following text as I_0, which we varied between experiments from 400 to 800 pA. On top of this constant current, we injected various other currents, usually sinusoidal at various frequencies and amplitudes. At relatively depolarized membrane potentials (i.e., more positive values of I₀), the cell fired to the sinusoidal current in tonic mode, whereas at relatively hyperpolarized membrane potentials (i.e., more negative values of I₀), the cell fired in burst mode; at intermediate levels of membrane potential, responses often consisted of a burst followed by tonic firing. It is noteworthy that in all cases when we saw both response modes to a cycle of the current injection, burst firing always preceded tonic firing.

General responses to current injection

In Fig. 1A, the results of current steps injected into the relay cell experimentally are shown. When the membrane potential (V_m) was adjusted initially to 58 mV (V_hold; I₀ = 230 pA), the neuron discharged in tonic mode in response to a short current pulse (200 pA). However, at a more hyperpolarized V_hold (77 mV), a depolarizing current step (50 pA) evoked a transient burst of high-frequency action potentials. The IFB model produces both tonic and burst responses (Fig. 1B) that are similar to those produced experimentally. Figure 1B also shows the inactivation gating variable of the simulated low-threshold Ca²⁺ current (denoted by h; see METHODS) dropping from unity to near zero shortly after the onset of the current pulse. In the IFB model, the time scale of this inactivation determines the length of the burst event. The value of h remains near zero (representing inactivation of the low-threshold Ca²⁺ conductance) until the depolarizing pulse ends, at which point the membrane potential drops below V_h, the threshold for deinactivation of I_T, and h recovers toward 1. 

View larger version (20K): [in this window] [in a new window]   Fig. 1. Experimental records of relay neuron responses to depolarizing current steps and comparable integrate-and-fire-or-burst (IFB) model simulations. A: relay neuron responds in either tonic or burst mode depending on the initial membrane potential (Vhold). For tonic (or burst) responses, Vhold in mV, initial applied current and applied current level during pulse in pA: 58, 230, and 430 (or 77, 350, 300). B: IFB model reproduces both tonic and burst responses depending on the initial membrane potential. Applied current initially and during pulse in µA/cm2: 0.67, 1.33 (0.16, 0.34). Bottom: h, the deinactivation gating variable for IT, during the IFB model burst response. Cellular parameters: gT = 0.8 mS/cm2; in mV: V = 50, Vreset = 60, Vh = 70, and VL = 75. Other parameters as in Table 1.

Figure 2A presents experimental recordings that illustrate firing patterns of a geniculate relay neuron during sinusoidal current injection over a range of stimulation amplitudes (I₁) while the modulation frequency is held constant at 1 Hz. In Fig. 2A, left, the neuron is in tonic mode, because of a more depolarized I₀ of 335 to 498 pA, whereas in the right column, the cell is hyperpolarized because of a more hyperpolarized I₀ of 4 to 136 pA and responds with burst discharges. When the neuron responds in tonic mode, the average number of spikes/cycle increases as I₁ is increased. In contrast, in burst mode, the neuron responds with 1 burst/cycle over a wide range of I₁, and the number of spikes/burst remains relatively constant at 7 or 8. An exception to this occurs at the lowest-modulation amplitudes, where the neuron bursts once for every several cycles of applied current (Fig. 2A, top right). In this paper, we will refer to such behavior as a subharmonic burst response. In addition to these representative patterns of cell responses to 1 Hz stimulation, we also observed superharmonic burst responses (2 bursts/cycle), subharmonic tonic responses, and burst followed by tonic responses.

View larger version (20K): [in this window] [in a new window]   Fig. 2. A: whole cell recordings of a single relay neuron showing representative firing patterns during 1-Hz sinusoidal current injection over a range of modulation amplitudes (I1). Tonic or burst responses are evoked depending on the value of the mean applied current (I0), which is 335-498 pA for tonic firing and 4 to 136 pA for burst firing. During tonic responses, the average number of spikes/cycle increases as I1 is increased in successive rows. Subharmonic burst responses are observed for low I1, but otherwise 1 burst/cycle is observed. B: behavior of the IFB model. When stimulus parameters are varied in qualitative agreement with A, the IFB model reproduces many salient features these responses. With depolarizing mean applied current (1.0 µA/cm2 < I0 < 1.2 µA/cm2), the IFB model responds in tonic mode and the average number of spikes/cycle increases as I1 is increased in successive rows. With hyperpolarizing mean applied current (I0 = 0.05 µA/cm2), the IFB model responds in burst mode. One burst/cycle is observed over a wide range of I1. In this figure and all that follow, cellular parameters are as in Table 1 unless otherwise noted.

Figure 2B presents a series of IFB model calculations for comparison with Fig. 2A. These calculations were performed using identical cellular parameters (see Table 1), whereas applied current parameters were chosen in qualitative agreement with the experimental conditions used in Fig. 2A. The simulations reproduce many salient features of the experimental recordings. For example, the IFB model responds in tonic mode to more depolarizing mean applied current (I₀) and in burst mode to more hyperpolarizing I₀. In tonic mode, the average number of spikes/cycle increases as a function of I₁, whereas in burst mode, 1 burst/cycle is observed over a wide range of I₁. In addition, the IFB model produces 7-8 spikes/burst, relatively independent of I₁. Although the IFB model reproduces the overall pattern of responses to a range of stimulus conditions in both tonic and burst responses, it does not reproduce the subharmonic responses observed at low modulation amplitude (cf. Fig. 2, A and B, top right; see DISCUSSION).

Figure 3A consists of recordings from the same neuron as Fig. 2A, but here sinusoidal current injection over a range of I₀ values and stimulation frequencies is applied while I₁ is fixed. When the holding V_m is adjusted so that the neuron is in a tonic firing mode (depolarized I₀; left), the cell exhibits tonic firing in response to I₁ frequencies of 0.3, 1, and 3 Hz, and is unresponsive to 10 Hz. The average number of spikes/cycle decreases in tonic mode as frequency is increased. However, when the neuron is in burst mode (hyperpolarized I₀; right), 1 burst/cycle (5-8 spikes/burst) is observed in response to 0.3 and 1 Hz until subharmonic responses are evoked at 3 Hz; no response was seen at 10 Hz. The subharmonic responses were observed most commonly in response to a frequency of 3 Hz, occasionally at 1 Hz. Many other cells did respond to 10 Hz stimulation, especially in tonic mode (see following text). In 4 of 12 cells, superharmonic responses of 2 bursts/cycle were observed at low stimulation frequencies (0.1-0.3 Hz; see following text).

View larger version (17K): [in this window] [in a new window]   Fig. 3. A: intracellular recordings from the same neuron as Fig. 2A showing responses to sinusoidal current injection over a range of frequencies (f) while the modulation amplitude is held constant (I1 = 96 pA). Relay neuron responds in tonic mode when a depolarizing mean applied current (I0 = 336 pA) is applied and responds in burst mode when I0 is hyperpolarizing (I0 = 136 pA). One burst/cycle is observed over a range of frequencies and subharmonic responses are observed as the cutoff frequency of 3 Hz is approached. B: IFB model responses that reproduce these firing patterns. IFB model responds in tonic mode when the mean applied current is depolarizing (I0 = 0.8 µA/cm2). Average number of spikes/cycle decreases as f is increased in successive rows; at 10 Hz, the model no longer responds with tonic spikes. IFB model responds in burst mode when the mean applied current is hyperpolarizing (I0 = 0.05 µA/cm2). One burst/cycle is observed over a wide range of f; bursts cease at f = 10 Hz. I1 = 0.2 µA/cm2.

Figure 3B presents a series of IFB model calculations that reproduce many salient features of the experimental results shown in Fig. 3A. For example, when the model responds in tonic mode (left), the average number of spikes per cycle decreases as frequency increases. Yet when the model responds in burst mode, 1 burst/cycle is observed over a wide range of frequencies of I₁. At the cutoff frequency of 10 Hz, both the relay neuron and the IFB model are unresponsive, and the effect of the mean applied current, I₀, on the mean membrane potential can be seen clearly (see Fig. 3, bottom). The IFB model thus qualitatively reproduces the neuron responses over a range of stimulus frequencies with the exception of subharmonic responses.

Superharmonic burst responses

As mentioned in the preceding text, burst responses observed at low frequency were primarily 1 burst/cycle (1:1). However, superharmonic burst responses were sometimes observed at low frequencies. Examples recorded from two geniculate relay cells of single (1:1) and double (2:1) burst responses at low frequency are shown in Fig. 4, A and B, respectively. Figure 4, left, shows responses to a more hyperpolarized I₀, and the right shows responses to a more depolarized I₀ (see legend for details). The result is pure burst responses on the left and burst followed by tonic responses on the right. Because superharmonic burst responses were seen in response to the lowest temporal frequencies tested, which was 0.1 Hz, the prevalence of these two response types in our data have been quantified in the following way. There were a total of 56 trials collected from 12 different cells that exhibited bursts at 1-3 Hz. Of these, we quantified the response type at the lowest frequency tested, 0.1 Hz. The majority (61%) were 1:1 (i.e., 1 burst/cycle as in Fig. 4A). Superharmonic bursts (i.e., 2:1 as in Fig. 4B) were observed in 14% of the trials, whereas 3 bursts/cycle (3:1) were never observed. Interestingly, in the remaining 25% of trials, no response at all was observed at 0.1 Hz. However, at a higher frequency (0.3 Hz), only 3 of 56 trials (5%) exhibited no response, indicating that burst mode for some neurons has a genuine band-pass character to varying temporal frequency (see following text).

View larger version (10K): [in this window] [in a new window]   Fig. 4. Representative responses for relay neurons driven at low frequencies. Asterisks indicate bursts. A: at 0.1 Hz, this relay neuron responds with 1:1 bursting (i.e., 1 burst/cycle). B: different relay neuron responds to 0.1-Hz stimulation with 2:1 superharmonic bursting (i.e., 2 bursts/cycle). In both cases, increasing I0, I1, or both leads to the recruitment of tonic spikes and the generation of a burst followed by tonic (burst-tonic) response. I0/I1 in pA for the burst and the burst followed by tonic firing case, respectively: A, 0/300, 130/300; B, 60/150, 170/240.

Responses as a function of frequency and amplitude of current injection

The filtering characteristics of relay neurons differ depending on the firing mode of the neuron. Figure 5 summarizes responses of two relay cells in both burst (hyperpolarized I₀) and tonic mode (depolarized I₀) to different levels of I₁ and frequency of current injection. The responses have been categorized into the following four classes: burst followed by tonic (filled circles), only burst (open circles, left) or only tonic (open circles, right), subharmonic burst (asterisks, left) or subharmonic tonic response (asterisks, right), and no response (dashes). Figure 5A, left, summarizes the responses of one of these neurons in burst mode. At low I₁ (50 pA), this neuron responded to 3 Hz with subharmonic bursts (asterisks) and gave no responses to higher or lower frequencies. With increasing I₁ (100 pA), the cell now responds 1:1 at 0.3 and 1 Hz, subharmonic at 3 Hz, but is still unresponsive to 0.1 Hz. With further increases in I₁ (200 pA), the neuron now responds to the full range of 0.1-3 Hz, but is unresponsive to 10 Hz. Thus burst responses of this cell show band-pass filtering for lower I₁ and low-pass filtering for higher I₁in the latter case, the high-frequency cutoff increases to 10 Hz for I₁ = 300pA. Similar results from a second neuron responding in burst mode are presented in the Fig. 5B, left.

View larger version (18K): [in this window] [in a new window]   Fig. 5. For a range of applied current modulation amplitude (I1) and frequency (f), the responses of 2 representative relay neurons, A and B, are categorized as burst followed by tonic firing (burst-tonic; filled circles), only burst or only tonic (open circles), subharmonic burst or subharmonic tonic (stars), and no response (dash). Superharmonic burst responses were not exhibited by either neuron. Burst followed by tonic responses also were observed at I1 values of 600 pA (not shown).

In tonic mode (Fig. 5, right), similar values of I₁ and frequency produce a different pattern of responses than observed in burst mode. First, all responses in tonic mode exhibit either low-pass or broadband filtering; there is no band-pass filtering because the cells always respond well to the lowest frequencies tested. Second, the high-frequency cutoff was greater in tonic than in burst mode, and for high values of I₁, there may be no observable cutoff frequency (Fig. 5A, top right). In this example, no high-frequency cutoff is observed because I₀ was greater than the rheobase of neuron, whereas in the other example (Fig. 5B, right), I₀ was less than the rheobase. It is also notable that subharmonics in tonic mode, when apparent, generally occurred before the cutoff frequency.

Fourier analysis of relay cell and IFB model responses

To compare quantitatively the different consequences of burst and tonic firing modes on relay cell responses to sinusoidal current injection, we performed Fourier analysis of the intracellular recordings (see METHODS). SPDHs were constructed from experimental recordings, and Fig. 6 shows examples of how this is done for several different stimulus conditions producing burst, tonic, or burst followed by tonic firing (see following text for details of stimulation parameters). Figure 6A shows responses to four cycles of the injected current. Figure 6B shows these responses aligned on a cycle-by-cycle basis, and Fig. 6C shows SPDHs constructed by assigning each spike to 1 of 64 bins depending on the value of its phase with respect to I₁. In the resulting histograms, the spike density, , approximates the likelihood of a neuron firing an action potential at a given phase, , of I_app. To quantify the dependence of the SPDHs on stimulus parameters (f, I₀, and I₁), discrete Fourier transforms of such histograms were performed leading to the assignment of four response measures: F₀, the mean firing rate; F₁, the stimulus- or modulation-driven component of the response; P₁, the phase advance or lag of the modulation-driven response; and , the nonlinearity of the response.

View larger version (22K): [in this window] [in a new window]   Fig. 6. Preliminary steps in the analysis of experimental data and simulation results. A: experimental voltage time courses (or simulation results) were obtained for several cycles of a stimulus with fixed parameters (f, I0, and I1). Here experimental voltage time courses that exhibit tonic, burst, and burst followed by tonic responses to stimulation at 0.3 Hz are presented. B: responses were aligned and each action potential was assigned a phase (0.5 <  <0.5) with respect to the sinusoidal applied current, phase zero being defined to occur when the applied current is maximum. C: spike phase density histogram (SPDH) was constructed by assigning each spike to 1 of 64 bins depending on the value of its phase. Resulting histogram represents the likelihood of a spike occurring at a given phase of the applied current. Discrete Fourier transforms were performed on these SPDHs leading to the calculation of the response measures, F0, F1, P1 and  (see METHODS).

Figure 6C shows examples of SPDHs that are representative of our results from relay cells exhibiting tonic, burst, and burst followed by tonic responses obtained at 0.3 Hz. When stimulus parameters were such that the neuron responded in tonic mode (I₀ = 410 pA and I₁ = 150 pA), the SPDH approximates the shape of a rectified cosine. When stimulus parameters were such that the neuron responded in burst mode (I₀ = 4 pA and I₁ = 50 pA), the SPDH does not approximate a rectified cosine but rather shows a sharp peak near  = 0.25 (i.e., a 90° phase advance). These features are combined in a SPDH generated from intracellular records that exhibited burst followed by tonic responses (Fig. 6C). The burst and tonic portions of this response are separated by ~10°, and because the burst always proceeds the tonic response of each cycle, there is a gap in the SPDH.

INTERPRETATION OF RESPONSE MEASURES. Because many of the results to follow involve the relationship between the response measures, F₀, F₁, P₁, and , it is instructive to review our expectations of these measures. Imagine an idealized tonic response of a neuron to be proportional to the rectified cosine (see Fig. 6C), most commonly

(6)

where R indicates the degree of rectification, H ( · ) is the Heaviside step function, and the maximum response is given by

(7)

Figure 7E shows examples of SPDHs of this form and Fig. 7, A and B, shows the resulting dependence of the response measures F₀ and F₁ (open circles with solid line) on the degree of rectification, R. As expected, both F₀ and F₁ are decreasing functions of R, because rectification decreases both the mean firing rate and modulation-driven component of the rectified cosine response. Figure 7C shows for tonic firing the quotient F₁/F₀ as a function of R. This plot indicates that a sinusoidal response with no rectification (and a mean equal to its modulation amplitude) will have F₁/F₀ = 1. Conversely, highly rectified responses will have F₁ approximately twice as large as F₀.

View larger version (20K): [in this window] [in a new window]   Fig. 7. Analytic calculation of response measures, F0, F1, and , for rectified cosine and square pulse responses given by Eqs. 6 and 8, respectively. A: solid line with open circles shows the normalized fundamental response as a function of the degree of rectification (R) when the poststimulus response histograms (PSTHs) are rectified cosines (see Eq. 6 and E for examples). Dotted line and solid line with open squares show the analogous calculation when the PSTHs are square pulses (see Eq. 8 and F). Solid (realizable) portion of the plot indicates the range of R that is experimentally observed during burst responses; dotted (unrealizable) portion indicates the range not observed (R < 0.85), due to the fact that a maintained burst response requires the stimulus to be hyperpolarizing during a significant fraction of the stimulus cycle. B-D: calculations similar to A for the fundamental response (F1), the ratio F1/F0, and the index of nonlinearity (), each plotted as a function of R for both the square pulse and rectified cosine cases. E and F: PSTHs for square pulse and rectified cosine responses that correspond to open circles and squares in A-D.

(8)

FOURIER ANALYSIS OF EXPERIMENTAL DATA. With this background, Figs. 8 and 9 present the results of Fourier analysis of the responses of a population of relay cells showing burst and tonic responses to sinusoidal current injection. Only data involving pure burst or tonic responses (i.e., no burst followed by tonic responses) are shown here, which means that these data reflect mostly small values of I₁ and/or extremely hyperpolarized or depolarized values of I₀. Figure 8, A-D, summarizes experimental results from 84 trials (8 different cells) in which only burst responses were observed. When plotted in units of spikes/second, the mean firing rate (F₀) is approximately half the value of the modulated response (F₁), and both are band-pass with peaks at 3 Hz (Fig. 8A). These data subsequently are plotted in units of spikes/cycle (Fig. 8B), showing that, at lower frequencies, a relatively constant number of spikes/cycle are observed, and in the majority of cases, these result from one burst/cycle.

View larger version (14K): [in this window] [in a new window]   Fig. 8. Fourier analysis of relay neuron burst responses to sinusoidal current injection. A-D: data from 84 trials (8 different cells) in which only burst responses were observed. Mean ± SD (in pA): I0 = 56 ± 112, I1 = 191 ± 14. Circles and bars indicate the mean ± SE for response measures (F0, F1, P1, and  ) as a function frequency (f). Mean firing rate (F0, open circles) and modulation-driven component of the response (F1, filled circles) are plotted first in units of spikes/s (A) and subsequently in units of spikes/cycle (B). Also shown as a function of temporal frequency are the phase of the modulated response component (C) and index of nonlinearity (D). E: SPDHs for a single neuron responding in burst mode to applied current of different frequencies. This neuron did not respond at 10 Hz. Three-hertz response is 1 burst/cycle.

View larger version (24K): [in this window] [in a new window]   Fig. 9. Fourier analysis of relay neuron tonic responses to sinusoidal current injection. A-D: data from 88 trials (10 different cells) in which only tonic responses were observed. Mean ± SD (in pA): I0 = 465 ± 133, I1 =193 ± 13. Conventions are identical to Fig. 9. F0 and F1 show no cutoff at high-frequency because I0 was often superthreshold. E: SPDHs for a single neuron responding in tonic mode to applied current of different frequencies. Superharmonic and subharmonic responses (S spikes every C cycles) are indicated by S:C.

FOURIER ANALYSIS OF THE IFB MODEL. Figures 10 and 11 present the results of Fourier analysis applied to IFB model responses. As in Figs. 8 and 9, I₀ and I₁ are chosen to ensure pure tonic or burst responses from the model (burst followed by tonic responses are considered in the following text). For burst firing, there is a remarkable, quantitative equivalence between experiment (Fig. 8) and theory (Fig. 10). One subtle difference is that the IFB model phase locks more precisely than actual relay cells (cf. 3-Hz SPDHs in Figs. 8E and 10E). This may be due to the presence of noise in the voltage recordings that is absent in the IFB model. Indeed, when the simulated applied current is supplemented with Gaussian noise (mean amplitude of 0 pA and a variance similar to that of I₁), phase locking by the IFB model is strongly attenuated (data not illustrated), and the nonlinearity index () is reduced. IFB model tonic responses (Fig. 11) are also comparable with experimental records (Fig. 9), though a notable exception is a gradual drop in  at 30-100 Hz that is seen experimentally but not reproduced (cf. Figs. 9D and 11D). Again, this difference may reflect the presence of noise in the experimental recordings because we would expect a uniform amount of spike jitter (in units of time) to be more apparent in SPDHs obtained during high-frequency stimulation (when bins of phase correspond to shorter time intervals). Because in the in vivo experimental condition there is more synaptic (and other sources of) noise than encountered in vitro, we would expect less phase locking and more linearity in vivo than we have seen experimentally here (Carandini et al. 1996). Nonetheless, phase-locked relay neuron responses to drifting sinusoidal contrast gratings have been observed in vivo (Reich et al. 1997, 1998).

View larger version (14K): [in this window] [in a new window]   Fig. 10. A-D: Fourier analysis of IFB model burst responses to sinusoidal current injection (cf. Fig. 8). I0 = 0 µA/cm2 and I1 = 0.33 µA/cm2. E: SPDHs for IFB model responding in burst mode to applied current of different frequencies.

View larger version (22K): [in this window] [in a new window]   Fig. 11. A-D: Fourier analysis of IFB model tonic responses to sinusoidal current injection (cf. Fig. 9). I0 = 1.11 µA/cm2 and I1 = 0.67 µA/cm2. E: SPDHs for IFB model responding in tonic mode to applied current of different frequencies. Superharmonic and subharmonic responses (S spikes for every C cycles) are indicated by S:C.

Phase plane portrait of the IFB model and high-frequency roll off in burst mode

As the cutoff frequency is approached, burst responses of relay cells gradually decline or roll off (e.g., Fig. 8, A and B). One obvious reason for this is that some neurons exhibit subharmonic bursts in this frequency range, and this would cause a decrease in spikes/second and spikes/cycle. In a subset of trials that exhibited bursting in the range of 1-3 Hz, we found subharmonic (1:N) bursting and 1:1 bursting to be nearly equally prevalent at 3 Hz (36 and 43%, respectively), suggesting for some cells another reason for this roll off, one that is predicted by our IFB model. Figure 12A shows burst responses from the IFB model that demonstrate the roll off in F₀ during 1:1 bursting. At 2 Hz, the model responds with 1 burst/cycle, and each burst is composed of 6 spikes/burst. However, at 6 Hz, where 1 burst/cycle also is seen, only 2 spikes/burst are evoked. Also note that the maximum deinactivation levels (h_max) of I_T achieved during the hyperpolarizing phase of the applied current is much greater at 2 than 6 Hz (Fig. 12B). The greater h_max at 2 Hz thus leads to a larger evoked low-threshold Ca²⁺ spike, which, in turn, evokes more action potentials.

View larger version (19K): [in this window] [in a new window]   Fig. 12. Frequency dependence of IT inactivation. A: membrane potential time course for IFB model at f of 2 (left) and 6 Hz (right). B: time course of h, the inactivation gate of IT, at 2 and 6 Hz. C: h vs. V phase-plane portraits for the IFB model simulations shown in A and B. Dashed line (labeled Vh), threshold for deinactivation (and activation) of IT; dot-dashed line (labeled Vreset), value to which the membrane potential is set after each spike; dotted line (labeled V), action potential threshold. Arrows, direction of flow; numbers make correspondence between C and A. D: frequency-dependence of hmax, the maximum deinactivation level achieved during repetitive bursting. E: number of spikes/burst for both experimental observations and the IFB model. Shown are 2 representative relay neurons (open squares and diamonds), the IFB model with standard parameters (open circles), and the IFB model with h+ = 300 ms (filled circles). I0 and I1 in µA/cm2: 0.0, 1.0.

Figure 12C presents phase-plane portraits of the IFB model at 2 and 6 Hz. This helps to clarify the relationship between membrane potential (V) and thus spike discharge, and the inactivation gating variable, h, at both frequencies. The solid line and arrows show the trajectory in the (V, h) plane that is repeated from cycle to cycle. The threshold for I_T (V_h, dashed line), V_reset (dot-dashed line), and V (dotted line) also are indicated. During the hyperpolarizing phase of the applied current, V eventually drops below V_h, causing h to increase (arrow 1). Eventually the current reverses, leading to depolarization; however, because V is still less than V_h, h continues to increase (arrow 2). When the membrane potential crosses V_h, I_T activates, h begins to drop, and I_T depolarizes the model neuron until the spike threshold, V, is reached (arrow 3). A series of action potentials are evoked (arrow 4) and h decreases until the sum of I_T and the applied current are no longer large enough to bring the membrane potential above threshold. When the applied current again reverses, the membrane potential hyperpolarizes, V eventually drops below V_h, and the periodic burst response repeats.

Figure 12D shows a plot of the frequency-dependence of h_max. Because the time constant for inactivation of I_T is smaller than the time constant for its deinactivation (see METHODS), h_max decreases as frequency increases. This, in turn, means that the size of the evoked low-threshold Ca²⁺ spike and the number of action potentials riding its crest will decrease at higher frequencies. It is thus this decline in h_max as a function of frequency that leads to the high-frequency roll off in the IFB model burst response. Although we do not have direct access the gating variable, h, in our experimental recordings, the open squares and diamonds in Fig. 12E show the number of spikes/burst exhibited by two relay neurons that burst 1:1 at all frequencies tested. This qualitatively matches roll off in spikes/burst exhibited by the IFB model (open circles) using standard parameters, although a better fit for these particular trials was obtained by increasing _h⁺ from 100 to 300 ms, which resulted in a lower cutoff frequency (filled circles).

Dependence of response measures on modulation amplitude

In the analyses summarized in Figs. 8-11, I₁ was fixed and small enough to avoid burst followed by tonic responses, and I₀ controlled the response mode by being either relatively depolarizing (for tonic firing) or hyperpolarizing (for burst firing). However, in both modes, the quantitative value of response measures depends on I₀ and I₁. This is true for both actual relay cells as well as the IFB model. Figure 13, A-C, left, presents the frequency dependence of the fundamental response F₁ for 11 relay neurons using either low I₁ (50-200 pA; filled squares) or high I₁ (300-500 pA; open triangles), and the right panels show the comparable responses from the IFB model. These results are pooled according to the firing mode based on I₀ so that Fig. 13A (I₀ high) presents responses that are predominantly tonic, whereas Fig. 13C (I₀ low) presents predominantly burst responses, and Fig. 13B (I₀ medium) includes many burst followed by tonic responses.

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