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Rate Coding and Spike-Time Variability in Cortical Neurons With Two Types of Threshold Dynamics

Department of Physiology, University of Cambridge, Cambridge, United Kingdom

Submitted 30 June 2005; accepted in final form 22 December 2005

	ABSTRACT

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
Neurons and dynamical models of spike generation display two different classes of threshold behavior: type 1 [firing frequency vs. current (f–I) relationship is continuous at threshold] and type 2 (discontinuous f–I). With steady current or conductance stimulation, regular-spiking (RS) pyramidal neurons and fast-spiking (FS) inhibitory interneurons in layer 2/3 of somatosensory cortex exhibit type 1 and type 2 threshold behaviors, respectively. We compared the postsynaptic firing variability of type 1 RS and type 2 FS cells, during naturalistic, fluctuating conductance input. In RS neurons, increasing the level of independently random, shunting inhibition caused a monotonic increase in spike reliability, whereas in FS interneurons, there was an optimum level of shunting inhibition for achieving the most reliable spike generation and the most precise spike-time encoding. This was observed over a range of different degrees of synchrony, or correlation, in the input. RS cells displayed a progressive rise in spike jitter during natural-like transient burst inputs, whereas for FS cells, jitter was mostly kept low. Furthermore, RS cells showed encoding of the input level in the spike shape, whereas FS cells did not. These differences between the two cell types are consistent with a role of RS neurons as rate-coding integrators, and a role of FS neurons as resonators controlling the coherence of synchronous firing.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
Recent studies have shown that mammalian cortical neurons recorded in vivo exhibit highly variable responses to repeated presentation of the same stimulus (Britten et al. 1993; Snowden et al. 1992; Tolhurst et al. 1983; Tomko and Crapper 1974). In vivo recordings of conductance in cortical pyramidal cells indicate that cortical networks are typically highly active and induce a large fluctuating background synaptic conductance (Azouz and Gray 1999; Destexhe et al. 2003). Background presynaptic activity is thus an important source of unreliability and variability in spike timing of cortical cells in vivo. In contrast, in vitro recordings show that spike generation in cortical neurons can be precise and reproducible in response to repeated injections of an identical fast-fluctuating current (Mainen and Sejnowski 1995; Nowak et al. 1997) or conductance (Harsch and Robinson 2000). Several modeling and experimental approaches have suggested that the high variability of firing in vivo may be explained by synchrony or correlation of inputs in time (Softky and Koch 1993) or by independent inhibition (Shadlen and Newsome 1998).

In a study of the responses of axons isolated from Carcinus maenas to various intensities of rectangular current stimuli, Hodgkin (1948) found that some axons showed a continuous transition from rest to regular firing at arbitrarily low firing frequencies as the stimulus intensity is increased, whereas others showed an abrupt onset of regular firing at a nonzero frequency. These types of behavior are now referred to as type 1 and type 2, respectively. In the mammalian cortex, there is increasing evidence that with steady current stimulation, regular-spiking (RS) pyramidal neurons show type 1 behavior and fast-spiking (FS) inhibitory interneurons show type 2 behavior (Erisir et al. 1999; Kawaguchi 1995; Tateno et al. 2004).

To shed more light on the consequences of the two types of threshold dynamics on postsynaptic encoding during more natural fluctuating input, we used the conductance injection technique (Robinson and Kawai 1993) to compare the input–output relationships and spike-timing variability of pyramidal cells and interneurons in layer 2/3 rat somatosensory cortical slices. By injecting trains of unitary conductance transients, which electrically mimic simultaneous fast excitatory [-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA)–type] and inhibitory [-aminobutyric acid type A (GABA_A)–type] inputs (Harsch and Robinson 2000), we first examined the firing frequency versus Poisson-input rate (f–r) curves, at several levels of random shunting inhibitory input. Second, using two statistics—the reliability of spike occurrence and the spike-timing jitter—we measured the firing variability of spike generation in the two types of threshold dynamics at several levels of inhibitory input and at different degrees of input synchrony or correlation. Third, we studied the sensitivity of firing variability to the relative timing of excitatory and inhibitory inputs. Finally, we discuss some possible functional roles of the two types of threshold dynamics, in cortical networks driven by fluctuating transient synaptic inputs.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
Slice preparation and recording

Transverse slices were prepared from somatosensory cortex of 15- to 21-day-old Wister rats using standard techniques (Sakmann and Stuart 1995). During slicing, tissue was kept in sodium-free solution that had the following composition (in mM): 254 sucrose, 2.5 KCl, 26 NaHCO₂, 10 glucose, 1.25 NaH₂PO₄, 2 CaCl₂, and 1 MgCl₂. Slices (300 µm thickness) were cut on a vibrating slicer (Microslicer DTK-3000, D.S.K., Kyoto, Japan) and kept in Ringer solution at room temperature for 2 h before recording. The Ringer solution contained (in mM):125 NaCl, 2.5 KCl, 25 NaHCO₂, 25 glucose, 1.25 NaH₂PO₄, 2 CaCl₂, and 1 MgCl₂. Both slicing and recording solutions were equilibrated with 95% O₂-5% CO₂ gas to a final pH of 7.4. Slices were viewed with an upright microscope (Olympus BW50WI, Olympus UK, London, UK) using infrared differential interference contrast optics. All experiments were performed at 34 ± 1°C. Whole cell patch-clamp recordings were made from the somas of neurons in layers 2/3. Putative RS cells were of pyramidal morphology, whereas putative FS cells were selected on the basis of a nonpyramidal shape and multipolar dendrites (Connors et al. 1982). During recording, the slices were perfused continuously with Ringer solution in which 10 µM bicuculline or gabazine (Sigma), 10 µM CNQX, and 10 µM AP5 (Tocris Cookson, Bristol, UK) were included to block most intrinsic synaptic conductances. Somatic patch-pipette recordings were made with a Multiclamp 700A amplifier (Axon Instruments, Foster City, CA) in current-clamp mode, correcting for prenulled liquid junction potential. Whole cell recording pipettes (3.9–4.3 M resistance, Clark GC150T-7.5) were filled with the standard intracellular solution: 105 mM K gluconate, 30 mM KCl, 10 mM HEPES, 10 mM phosphocreatine Na₂, 4 mM ATP-Mg, and 0.3 mM Na-GTP, balanced to pH 7.3 with NaOH. Series resistance compensation was used. Signals were filtered at 5 kHz and sampled with 12-bit resolution at 20 kHz.

Conductance injection

For conductance injection (dynamic clamp) stimulation (Robinson and Kawai 1993; Sharp et al. 1993), an SM-1 conductance injection amplifier (Cambridge Conductance, Cambridge, UK) was used. The opening of a population of receptor channels at synapses is modeled by an excitatory (AMPA) receptor synaptic conductance g_E(t) and an inhibitory (GABA_A) receptor synaptic conductance g_I(t). Depending on the changing membrane voltage V(t), an injected current is described by

(1)

where E_E and E_I are the reversal potentials for the AMPA- and GABA_A-type conductances, respectively. For each cell, E_I was set at the membrane resting potential (Connors et al. 1988) and E_E = 0 mV (Hollmann and Heinemann 1994; MacDermott and Dale 1987).

Conductance stimuli

For two types of receptor conductances, the waveform of a single synaptic input was specified by the following function of time t

(2)

where j represents E or I, C_j is the scaling factor, _j^a is the activation time constant, and _j^d is the decay time constant (Harsch and Robinson 2000; Koch 1999; Sacchi et al. 1998). The scaling factor values were as follows (pS): C_E = 1,000 (Hestrin 1993; Stern et al. 1992; Stevens and Zador 1998) and C_I = 300 (Edwards et al. 1990; Ling and Benardo 1999; Salin and Prince 1996). The time constants were based on voltage-clamp measurements in the literature, adjusting for temperature differences. The time constant values were as follows (ms): _E^a = 0.5, _E^d = 2 (Häusser and Roth 1997; Kleppe and Robinson 1999), _I^a = 0.5, and _I^d = 7 (Kapur et al. 1997; Ling and Benardo 1999; Pearce 1993). For comparison, all the time constant values were the same as used in Harsch and Robinson (2000). The two channels of conductance stimulation were constructed by summing the unitary conductance transients such that

(3)

where j denotes E or I and, for k = 1, 2, ... , {T_k^j} is a point process of initiating synaptic unitary events.

Point processes

Once the parameters of the unitary synaptic input are fixed, the time series {T_k^j} (j = E or I, k = 1, 2, ...) completely determine the time course of conductance stimuli. As reported before (Harsch and Robinson 2000), we used two point-process models: 1) Poisson and 2) doubly stochastic process models. Poisson stimulus trains were constructed by summing unitary events, such as AMPA- or GABA-type, at intervals denoted by a random variable t_i, with the probability density

(4)

The intensity of stimulation was varied by changing the Poisson rate . The relationship between firing rate (f) and the excitatory input Poisson rate (_E) is afterward referred to as an f–r curve. To simulate correlated or synchronous firing of synaptic events, in addition, we used a doubly stochastic process model that mimics bursty presynaptic events. The primary Poisson process with a rate _b produces a nonstaionary secondary Poisson process, whose rate was modulated in a summation of exponentially decaying transients with a constant _b; the time-dependent rate (t) of the secondary process was described by

(5)

where S is an initial peak rate, {_k} is a time series of the primary events, and

(6)

The mean rate of synaptic events is then given by = S_bb. The intervals between synchronous bursts are random and exponentially distributed. To systematically vary the degree of synchrony or correlation in the input, we keep and _b fixed, and vary S and _b reciprocally, defining S, the amplitude of bursts in input rate, as the level of synchrony (see Fig. 5B). In practice, values of = 200 Hz and _b = 2.5 Hz were used for both excitatory and inhibitory conductance inputs when investigating the effect of synchrony on spike variability, and the ratio of values between excitation and inhibition was changed to investigate the effect of excitatory: inhibitory balance on spike timing (_I/_E = 0 to 1.2, in steps of 0.1 or 0.2). Similar doubly stochastic process models have been used to analyze bursting activity of lateral superior olive neurons (Turcott et al. 1994) and dissociated cortical neurons (Tateno et al. 2002). Test stimuli lasted for 2 or 5 s, and 30 s of recovery time at the resting potential were allowed between successive stimuli. To study the effect of relative timing of inhibition and excitation, GABA_A-type inhibitory conductance input g_I(t) was produced by advancing or retarding the AMPA-type excitatory Poisson stimulus g_E(t) in 1-ms increments; i.e., g_I(t) = g_E(t-delay) and delay = 0, ±1, ±2, ±3, ... (in ms).

View larger version (32K): [in this window] [in a new window]   FIG. 5. Doubly stochastic process (DSP) stimulus and evoked responses. A: schematic representation of the DSP model used to produce correlated or synchronized input timing. See METHODS for details. a: primary process with three parameters (S, b, and b). b: inhomogeneous Poisson point process. c: conductance input resulting from convolution of the point process and unitary synaptic event waveforms. B, a: response (top) to simultaneous, synchronous excitatory conductance input (middle), and inhibitory conductance input (bottom). For both excitatory and inhibitory conductance inputs, parameters: S = 200, b = 2.5, and b = 0.4. b: S = 500, b = 2.5, and b = 0.16. c: S = 800, b = 2.5, and b = 0.10.

Spike times were measured as the times of upward zero crossing of the membrane potential. Instantaneous frequency (reciprocal of each interspike interval) was computed from trains of action potentials evoked by 600-ms-duration pulses for the first, second, fourth, and last interspike intervals. Steady-state (SS) firing frequency was computed as the average of instantaneous frequency for the last three intervals of a train. Current or conductance strength was usually progressively increased or decreased in small (10- or 20-pA) steps. Initial instantaneous frequency and steady-state firing rate were plotted as a function of the injected current strength, to construct frequency–current (f–I) relationships. The maximum firing rate of a neuron was computed from the number of spikes per trial at the highest current strength before depolarization block. The frequency adaptation properties of neurons were characterized by calculating the instantaneous firing rate as a function of time since the beginning of the 600-ms pulse. For each current intensity, the decay of firing rate was fitted to a single exponential function

(7)

where f and t respectively represent the firing rate and time after the stimulus onset and C_A, _A, and F_A are positive constant parameters. F_A represents the adapted firing rate. The strength of adaptation (adaptation index, A) was quantified as 100 x (1 – F_A/F₁), where F₁ corresponds to the firing rate of the first interspike interval. Because adaptation depended on the current intensity for any given neuron, we used the highest current level not producing depolarization block of spiking, to allow comparison among cells. For some cells, no adequate exponential fit could be obtained and, in these cases, F_A was calculated as the mean firing rate for the last 50 ms of the 600-ms current pulse and used to calculate the adaptation index. Results are reported as means ± SD. Membrane time constants were obtained by fitting a single exponential function to the initial part of more than 10 time-averaged voltage responses to small (–20 or –10 pA), 600-ms-long hyperpolarizing current pulses. Input resistance was calculated from Ohm's law by dividing the maximal average voltage deflection by the amplitude of the applied current pulses. Firing frequency (f) versus excitatory Poisson rate (f–r) curves were fitted with the function

(8)

derived from a leaky integrate-and-fire model, where r is the excitatory Poisson firing rate in kilohertz, and ₀, , a, b, and c are all constant parameters (Koch 1999). The parameter ₀ determines the slope of the initial rise and determines the curvature and saturating value (i.e., the nonlinearity); the smaller the value of , the stronger the nonlinearity, although the nonlinearity depends on the parameter ₀ as well. All the parameters were estimated by minimizing mean-squared error between the model and the data. We report the values of ₀ and , which are the only parameters of interest here.

Data analysis

To calculate the firing statistics, the first 200 ms of responses were excluded to avoid the influence of adaptation. To characterize overall response variability, a statistical measure, the coefficient of variation (CV) of the interspike intervals, was used on ensembles of 20–35 successive trials, either 2 or 5 s in duration. CV = 1 for a stationary Poisson process. Each set of trials was termed a "session." For each session of trials, the stimulus was resynthesized with new random numbers (i.e., different initial random seeds).

To characterize reproducibility of spike times, two additional measures, termed "jitter" and "reliability," were used. For each session of 20–35 trials as described above, a peristimulus time histogram was constructed with a bin width of 1, 2, or 3 ms. The first 200 ms of responses were excluded to avoid the influence of adaptation. Repeatable spikes were defined by events in which firing occurred in the same time bin for 30% of trials and included spikes in these and the immediately adjacent time bins. Reliability was defined as the proportion of all spikes that were repeatable. Jitter was defined as the average value of the SD of spike times within each repeatable event. These measures of reproducibility are similar to those used by Mainen and Sejnowski (1995) and Nowak et al. (1997). Although varying the bin width shifted graphs of reliability and jitter values upward or downward, the bin width was not a critical factor in determining overall trends. Thus the results reported here use a bin width of 2 ms.

To analyze spike-timing variability in more detail during a decaying synaptic conductance, we also performed a finer analysis. In particular, we computed "individual" jitter and reliability values for the last three spikes during evoked burst responses. To find boundaries between two events, we used a discriminant analysis assuming that spike times in each event are normally distributed. In other words, if the mean spike times in events i and (i + 1) are t_i and t_i+1, respectively, and their SDs (individual jitters) are _i and _i+1, then the boundary between the two events is defined as (_i+1t_i + _it_i+1)/(_i + _i+1). In practice, once the boundary is determined, the elements belonging to each event may be rearranged. Thus we repeated this procedure iteratively until no further change occurred.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
Cell types in the layer 2/3 of rat somatosensory cortex

Based on responses to injected step currents, cells recorded in layer 2 or 3 of somatosensory cortex were classified into three groups: regular-spiking (RS), fast-spiking (FS), and low-threshold–spiking (LTS) cells (Beierlein et al. 2003; Connors and Gutnick 1990; Kawaguchi and Kubota 1997). This study is based on recordings from 40 RS and 43 FS cells. Because we recorded only a small number of LTS neurons (seven cells), the analysis of firing variability below will be reported only for RS and FS cells. As reported previously (Tateno et al. 2004), we selected RS cells that had a typical pyramidal morphology under infrared differential interference contrast optics, whereas FS cells were selected on the basis of a nonpyramidal shape, with a round soma and multipolar dendrites. Figure 1, A and B shows typical action potential waveforms for an RS cell and an FS cell, respectively, at three levels of injected step current. RS cells had an average resting membrane potential of –72.5 ± 4.5 mV, input resistance of 419 ± 181 M, membrane time constant of 39.2 ± 13.5 ms, maximum firing rate of 37.8 ± 8.5 Hz, adaptation index of 72.4 ± 8.9%, and adaptation decay time constant of 31.4 ± 10.4 ms. FS cells had an average resting membrane potential of –72.2 ± 3.5 mV, input resistance of 353 ± 143 M, membrane time constant of 29.1 ± 5.8 ms, maximum firing rate of 91.0 ± 11.1 Hz, adaptation index of 40.2 ± 6.9%, and adaptation decay time constant of 57.2 ± 31.5 ms. These statistics are summarized in Table 1. To distinguish RS and FS cells more reliably, we also used several other measures as reported in Tateno et al. (2004). In addition, it is notable that RS and FS cells characteristically have "type 1" and "type 2" threshold dynamics, respectively (see Tateno et al. 2004).

View larger version (32K): [in this window] [in a new window]   FIG. 1. Firing properties of two classes of neurons in layer 2/3 somatosensory cortex. A: repetitive firing of regular-spiking (RS) cells for 3 different current steps of increasing amplitude (90–380 pA). B: repetitive firing of fast-spiking (FS) cells for 3 different current steps of increasing amplitude (70–300 pA). Repetitive firing showed an abrupt onset in FS cells (bottom), starting at a critical frequency (about 20 spikes/s) in response to a 70-pA current step. Suprathreshold current injection produced repetitive firing at higher frequencies and with much less spike frequency adaptation than observed for RS cells (cf. A). FS cells had larger afterhyperpolarizations than RS cells.

To examine the effects of shunting inhibitory input on postsynaptic firing rate and interspike interval, cells were stimulated using the conductance injection technique (Robinson and Kawai 1993). Inhibitory GABA_A-type and excitatory AMPA-type Poisson conductance stimuli were synthesized and simultaneously injected to cells (see METHODS).

Figure 2Aa shows average firing frequency versus AMPA-type Poisson rate (f–r) relationships for an RS cell with four levels [0 (control), 1, 2, and 3 kHz] of GABA_A-type Poisson conductance. As shown in the figure, for RS cells, f–r curves are nonlinear and inhibitory input shifts f–r curves to the right. For the control (zero inhibition), the parameter ₀ (see METHODS, Eq. 8) was estimated as 2.4 ± 1.0 x 10^–2 ms and as 0.92 ± 0.31 ms. With inhibitory Poisson input at a rate of 2 kHz, ₀ was 2.9 ± 1.6 x 10^–2 ms and was 3.0 ± 1.1 ms. Another common finding in the recordings was that f–r curves for several different levels of inhibition crossed at higher excitatory input rates, presumably because inhibitory inputs release action-potential depolarization block at higher levels of excitatory inputs, weakening the nonlinearity and increasing the maximum firing frequency. For the RS cell of Fig. 2Aa, the coefficient of variation (CV) of interspike intervals at the three levels of inhibitory input and in the control condition is shown in Fig. 2Ab. Above about 3 kHz, CV converges to around 0.45 and is similar for all levels of inhibition.

View larger version (37K): [in this window] [in a new window]   FIG. 2. Firing statistics during simultaneous excitatory (AMPA-type) and inhibitory (GABA-type) Poisson conductance inputs. A: an RS cell. a: firing frequency vs. excitatory Poisson rate (f–r) relationship at 3 different levels (1.0, 2.0, and 3.0 kHz) of inhibitory Poisson conductance input. Control is without inhibitory conductance input. Addition of shunting conductance produces a rightward shift of f–r relationships. b: coefficient of variation (CV) as a function of excitatory Poisson rate at the three different levels of inhibitory Poisson input for the same cell in Aa. B: an FS cell. a: f–r relationship at 4 different levels (0.5, 1.0, 1.5, and 2.0 kHz) of inhibitory Poisson input. Control is without inhibitory input. Similarly, addition of shunting conductance produces a rightward shift of f–r relationships. b: CV as a function of excitatory Poisson rate at the 4 different levels of inhibitory Poisson input, for the same cell as in Ba.

Figure 3Aa shows averaged action-potential shapes in an RS cell at four rates of Poisson excitatory conductance input (2, 3, 4, and 5 kHz). The average spike properties such as spike width and amplitude strongly depended on the input rate as shown in Fig. 3Aa (see also de Polavieja et al. 2005). Spike-triggered averaging of the conductance stimulus (also known as "reverse correlation") shows that for RS cells, spikes are typically associated with a rise in AMPA conductance lasting 2–8 ms and peak about 3–4 ms before the spike as shown in Fig. 3Ba, as also reported in Harsch and Robinson (2000). During simultaneous inhibition, the form of the average excitatory conductance depended on the excitation rate, with a progressively more pronounced minimum preceding the peak, as shown in Fig. 3, Bb and Bc. A slower (5- to 15-ms) spike-associated depression of inhibitory conductance was also observed as shown in Fig. 3, Cb and Cc, which was much more prominent at low excitation rates.

View larger version (42K): [in this window] [in a new window]   FIG. 3. For an RS cell, average action potentials and average excitatory and inhibitory conductance changes associated with spikes. A: for 4–5 levels of excitatory conductance input, average action potentials in the absence (1st row) and presence of inhibitory conductance input (2nd row, 1 kHz; 3rd row, 2 kHz). B: for 4–5 levels of excitatory conductance input, average excitatory conductance changes associated with spikes. C: average inhibitory conductance changes.

View larger version (39K): [in this window] [in a new window]   FIG. 4. For an FS cell, average action potentials and average excitatory and inhibitory conductance changes associated with spikes. A: for 4–5 levels of excitatory conductance input, average action potentials in the absence (1st row) and presence of inhibitory conductance input (2nd row, 1 kHz; 3rd row, 2 kHz). B: for 4–5 levels of excitatory conductance input, average excitatory conductance changes associated with spikes. C: average inhibitory conductance changes.

Because firing of cortical neurons is typically correlated with population activity in the network, we examined how the clustering of conductance transients in time affects the measures of firing variability and the temporal fine structure of responses. For this purpose, we used a doubly stochastic process (DSP) model, composed of primary and secondary processes, to determine the input events (see METHODS). Briefly, the rate of the primary process in the DSP model is modulated in exponentially decaying transients (peak amplitude S and decay time constant _b), at intervals specified by a stationary Poisson process (rate _b) as shown in Fig. 5A. This allowed the degree of correlation in time to be varied, for any given _b, from an effectively stationary Poisson process to a highly clustered or synchronous input. This is evident in Fig. 5, Ba (S = 200) to Bc (S = 800).

To quantify the variability of postsynaptic spike timing evoked by conductance stimuli, we used two additional statistical measures: spike jitter and reliability (see METHODS). Although the CV describes the overall statistics of spike timing, it gives no insight into the reproducibility of responses to the same input. Therefore we examined spike timing in ensembles of responses to the same input, using the measures of spike jitter and reliability.

In an RS cell, at a specific level of inhibition (_GABA/_AMPA = 0.6), increasing synchrony decreased the firing rate in an approximately linear fashion as shown in Fig. 6Aa. As expected, greater synchrony increased CV of interspike intervals for the same cell (Fig. 6Ab). Figure 6, Ac and Ad shows that reliability and jitter both increased with the level of synchrony. FS cells (Fig. 6, Ba–Bd) showed a broadly similar pattern to RS cells in these measures, as a function of the synchrony parameter S, except that firing frequency increased rather than decreased with increasing S and jitter showed no clear trend with S. Table 3 summarizes the results for eight RS and nine FS cells that we analyzed.

View larger version (28K): [in this window] [in a new window]   FIG. 6. Dependency of firing statistics on synchrony in RS and FS cells. For an RS cell and an FS interneuron, 5 statistical measures to characterize firing properties and spike-time variability. A: RS cell at a specific ratio (GABA/AMPA = 0.6) between excitatory (AMPA-type) and inhibitory (GABA-type) conductance inputs. a: firing frequency vs. synchrony (S) or correlation in time. b: coefficient of variation (CV) of interspike intervals vs. synchrony. c: reliability vs. synchrony. d: spike jitter vs. synchrony. B: FS cell. a: firing frequency vs. synchrony (S) or correlation in time. b: coefficient of variation (CV) of interspike intervals vs. synchrony. c: reliability vs. synchrony. d: spike jitter vs. synchrony.

It is well established that CV measured in vivo is typically >1 (Gershon et al. 1998; Softky and Koch 1993). In contrast, for both RS and FS cells, CV during excitation by Poisson AMPA-type input is much lower than that in in vivo firing as seen in the control cases of Fig. 2, Ab and Bb. However, simultaneous independent inhibition can increase spike-time variability by adding variance to the input (Fig. 2, Ab and Bb). We investigated this effect in more detail, at high (Fig. 5Bc, S = 800) and low (Fig. 5Ba, S = 200) levels of correlation in the excitatory input.

In RS cells, independent random inhibition caused spikes to drop out, decreasing the firing rate at both low (S = 200, Fig. 7Aa) and high (S = 800, Fig. 7Ba) synchrony in the excitatory input. Setting the initial firing rate for _GABA = 0 to be 13–20 spikes/s in this experiment, we found that increasing the _GABA/_AMPA ratio reduced firing frequency in an approximately linear fashion for both levels of synchrony. Inhibition decreased CV of interspike intervals at high synchrony (S = 800, Fig. 7Bb), whereas it increased CV at low synchrony (S = 200, Fig. 7Ab). Reliability (Fig. 7, Ac and Bc) and jitter (Fig. 7, Ad and Bd) increased as the proportion of the inhibition was increased. Results for all eight RS cells analyzed are given in Table 4.

View larger version (20K): [in this window] [in a new window]   FIG. 7. Effects of balance between excitatory (AMPA-type) and inhibitory (GABA-type) conductance inputs on firing statistics in an RS cell. A: at a low level of correlation in time or synchrony (S = 200). a: firing frequency vs. inhibitory ratio to excitatory input. b: coefficient of variation (CV) of interspike intervals vs. inhibitory ratio. c: reliability vs. inhibitory ratio. d: spike jitter vs. inhibitory ratio. B: at a high level of synchrony (S = 800). a: firing frequency vs. inhibitory ratio to excitatory input. b: coefficient of variation (CV) of interspike intervals vs. inhibitory ratio. c: reliability vs. inhibitory ratio. d: spike jitter vs. inhibitory ratio.

View larger version (28K): [in this window] [in a new window]   FIG. 8. Optimum balance between inhibition and excitation for spike reliability and jitter in FS neurons. A: at a low level of synchrony (S = 200). a: for 3 different cells, reliability vs. inhibitory ratio to excitatory input. b: spike jitter vs. inhibitory ratio. c and d: histograms of ratios for maximum reliability (c) and minimum jitter (d) in 10 FS cells, cell-count histogram for the ratios of maximum reliability. Ba–d, as in A but at a high level of synchrony (S = 800).

Characteristic differences in the responses of RS and FS cells can be expected to emerge when they are stimulated by inputs close to their spike thresholds, arising from the difference of their threshold dynamics. We examined the consequences of this in responses of the two cell types to decaying synaptic input bursts, which pass through threshold with natural-like conductance fluctuations. As seen in the raster plots of Fig. 9, Ac and Bc, spike times become considerably less precise with time during the response as a result of intrinsic noise in the spike-generating mechanism of the neuron. To quantify the irregularity, we used jitter and reliability of individual events rather than the overall averages (see METHODS). Figure 9, Ae and Be shows jitter as a function of spike time occurrence for RS and FS cells, respectively. In both cell types, jitter increases rapidly over the first 200 ms, approximately. The final stage is quite different between RS and FS cells. Whereas in RS cells jitter keeps rising (except for the final spike), in FS cells jitter consistently levels off or is even reduced, over the majority of the response. This effect is seen more clearly in jitter versus reliability plots for the last three spikes over all sessions (Fig. 10). Points in RS cells scattered over the jitter–reliability plane (Fig. 10A), whereas those in FS cells concentrated in the left side area (i.e., jitter <4 ms; Fig. 10B). In other words, in comparison to RS cells, although FS cells are not necessarily reliable in producing spikes, the timing of spikes that are produced tends to be reliable.

View larger version (44K): [in this window] [in a new window]   FIG. 9. Response to a synchronous burst of simultaneous AMPA-type and GABA-type conductance input in RS and FS cells. A: RS cell. a: an example membrane potential response at the soma. b: conductance inputs consisting of a train of unitary AMPA-type and GABA-type conductance transients, generated by a nonstationary Poisson process with an exponentially declining rate. AMPA-type and GABA-type conductance inputs are shown, respectively. Parameters: S = 800 (Hz) and b = 0.313 (s). c: raster display of 25 trials with the identical stimulus. d: peristimulus time histogram with 2-ms bin width. Events were discriminated by thresholding at 8 spikes per bin. e: SD against mean of successive spike times for each event. f: superimposed membrane potential trajectories for spikes of different precisions, as indicated in a. B: FS cell. As in A, but using a session of 28 trials, and a threshold of 9 spikes per bin. e: SD against mean of spike times for each event. f: Superimposed membrane potential trajectories for spikes of different precision, as indicated in a.

View larger version (17K): [in this window] [in a new window]   FIG. 10. Distribution of jitter and reliability in RS and FS cells for the final spikes of transient responses. A: for 14 RS cells, jitter and reliability of last 3 spike events are shown for 58 sessions. Last 3 spike events in each trial, i.e., events N, N – 1, and N – 2 are indicated respectively by filled circles (), triangles (), and crosses (+). B: 117 jitter-reliability pairs over 39 sessions are shown, from 11 FS cells. See also Table 6.

Given the highly recurrent nature of the cortical columnar circuitry, a close temporal correlation among excitatory and inhibitory input to RS and FS cells is quite likely, and thus it is of interest to ask how spike-time reliability depends on the relative timing of inhibition and excitation. We examined jitter and reliability at a variety of delays between identically timed Poisson trains of inhibitory and excitatory events (Fig. 11A). In both types, results were broadly similar. Reliability showed a clear maximum when inhibition followed excitation by a delay of 2–3 ms as illustrated in Fig. 11, Ba and Ca, whereas spike timing jitter went through a clear minimum at about the same delay (Fig. 11, Bb and Cb). Histograms of optimal delay in 13 RS cells are shown in Fig. 11, Bc and Bd for reliability and jitter, respectively. The optimal delay was 3.0 ± 1.1 ms for reliability and 2.9 ± 1.2 ms for jitter. In 14 FS cells, the optimal delay was 2.4 ± 1.2 ms for reliability (Fig. 11Cc) and 2.7 ± 1.1 ms for jitter (Fig. 11Cd).

View larger version (25K): [in this window] [in a new window]   FIG. 11. Effects of delay between excitatory and inhibitory conductance inputs on spike-time variability. A, a: schematic representation of excitatory and inhibitory conductance stimuli with delay. b: response to the compound stimulation. B: RS cells: a: for 3 different cells, reliability vs. delay. b: spike jitter vs. delay. c and d: for total of 13 cells, histograms of the delays for (c) maximum reliability and (d) minimum jitter. C: FS cells: a: for 3 different cells, reliability vs. delay. b: spike jitter vs. delay. c and d: for a total of 14 cells, histograms of the delays for (c) maximum reliability and (d) minimum spike jitter.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

Recently, the conductance injection or dynamic clamp technique has been increasingly used to understand spike generation during electrically realistic conductance input that mimics natural synaptic input (Chance et al. 2002; de Polavieja et al. 2005; Mitchell and Silver 2003; Suter and Jaeger 2004; for review, Destexhe et al. 2003). This approach is ideal for investigating reliability and spike-time variability because a complex fluctuating conductance input can be applied repeatedly and exactly. Here, our interest focuses especially on how the firing rate and spike-time variability can be modulated by simultaneous excitatory and inhibitory conductance stimulation in RS and FS cells, with their different types of threshold dynamics.

The firing frequency versus Poisson-input rate (f–r) curves at several levels of fluctuating inhibitory input showed that, in both RS and FS cells, random background inhibitory input shifted the f–r relationships rightward. This simple shift was independent of the type of threshold dynamics and differs from the multiplicative gain modulation by inhibition described in Chance et al. (2002) and Mitchell and Silver (2003), perhaps because of less dependency of the subthreshold membrane potential on the level of excitation in these cells and/or less difference from the reversal potential of inhibition. Furthermore, the extent of nonlinearity and the amount of the shift and linearization by inhibition differed between the two classes of cells (Fig. 2, Aa and Ba), with RS cells showing a depolarization block at high excitatory input rates, which could be relieved by background inhibition.

It is well appreciated that in a variety of in vitro preparations, stimuli with fast fluctuations resembling synaptic activity in vivo produce more reliable firing (Bryant and Segundo 1976; Harsch and Robinson 2000; Mainen and Sejnowski 1995; Nowak et al. 1997; Suter and Jaeger 2004; Tang et al. 1997). To compare with previous work, we used five statistical measures of spike variability averaged over hundreds of spikes arising from different trajectories of membrane potentials in different stimulation conditions.

In this study both RS and FS cells showed much less variability of spike intervals in response to steady Poisson AMPA excitation than that in vivo, where CV >1 (Buracas et al. 1998; Stevens and Zador 1998). In vivo, however, clusters of spikes or bursts are driven by concerted activity in multiple nearby cells (Azouz and Gray 1999; Tsodyks et al. 1999) and bursting plays a significant role in sensory information transmission (Krahe and Gabbiani 2004). Therefore we introduced correlation in time or synchrony into excitatory and inhibitory conductance stimuli, using a nonstationary Poisson process as described (Harsch and Robinson 2000).

We found, in agreement with this previous study, that for RS cells, synchronous stimulation increased variability for the same mean input rate, and made it easy to increase variability to the in vivo levels, although we did not use N-methyl-D-aspartate (NMDA)–type conductance in this study. The novel aspect of this study was to extend these results to FS cells. Variability and spike-time reliability increased in both RS and FS cells with increasing synchrony. The doubly stochastic input increased variability of spike trains, which exhibits two different time scales: within and between bursts. By changing the duration of clusters relative to their rate, we were able to systematically vary the degree of synchrony in the input. A linear relationship was found between each variability measure and the degree of synchrony in both types of single cells. This result indicates that variability is in part determined by input synchrony in RS and FS cells and that spike clusters or bursts may be more reliable codes in both cell types than are single spikes.

The high in vivo level of spiking variability of spiking has also been suggested to result from independent random inhibitory input in cortical neurons (Shadlen and Newsom 1998) and in cerebellar Purkinje cells and interneurons (Häusser and Clark 1997). However, in cortical neurons, injection of neither random inhibitory current (Stevens and Zador 1998) nor conductance (Harsch and Robinson 2000) was able to reproduce in vivo levels of variability. Here, we found that, in both RS and FS cells, increasing the level of inhibition had opposite effects on CV, depending on the level of synchrony; at low synchrony or with stationary Poisson stimulation, inhibition increased CV (Figs. 2, Ab and Bb, 7Ab, and 8Ab, S = 200), whereas at high synchrony, it decreased CV (Figs. 7Bb and 8Bb, S = 800). This supports the conclusion that high in vivo CV values do not solely arise from the additional variance of random inhibition.

For FS neurons, there appears to be an optimum level of shunting inhibition for achieving the most precise spike-time encoding. This was observed over a range of different degrees of synchrony in the excitatory and inhibitory input. FS neurons have an intrinsic lower limit of stable firing frequency at around 20–30 Hz ("type 2" threshold dynamics) as well as subthreshold oscillations at the similar frequency, whereas RS neurons do not ("type 1") (Tateno et al. 2004). This optimization of FS cell firing precision might arise when the subthreshold membrane potential dynamics are tuned by the level of shunting inhibition so as to maximize subthreshold oscillation amplitude. This could have functional relevance in controlling the coherence of synchronous firing, which is believed to depend critically on the FS cell network.

In the cortex, RS and FS cells receive common thalamic input and FS cells inhibit local RS cells. It can thus be expected that many RS cells should receive inhibition whose timing is tightly coupled to excitation as, for example, reported in auditory cortex by Wehr and Zador (2003). Here, we showed that with identically timed inhibition and excitation, a delay of 2–3 ms between excitation and inhibition produces optimal reliability and jitter. This essentially occurs because inhibition restricts the generation of spikes to brief time windows during stronger excitatory fluctuations and the effect is independent of threshold dynamical type.

In the neocortex, pyramidal RS neurons collect widely distributed inputs delivered in different layers and make excitatory synapses to other neurons. Because of relatively strong spike adaptation and type 1 threshold dynamics, the coherence of RS cell firing should fail relatively easily at weaker levels of input and during the decaying phase of transient inputs. Thus RS cells promote spike-time variability and are, in this sense, suited to rate coding and the transmission of transient information, rather than temporal coding and the maintenance of coherent rhythms (see Figs. 9A and 10A). In contrast, inhibitory interneurons of the same type (FS or LTS cells) are strongly interconnected not only by electrical synapses or gap junctions but also by chemical synapses, and are implicated in promoting synchronous firing (Beierlein et al. 2000; Galarreta and Hestrin 2001; Gibson et al. 1999). In addition, FS interneurons have type 2 threshold dynamics and little spike adaptation. The existence of subthreshold oscillations in FS cells means that the phase of rhythmic synchronous firing among FS cells can be kept stable even if input drops below the threshold level. Although, by virtue of their discontinuous f–I relationship, FS cells have a harder onset of spike initiation and are consequently less reliable in producing spikes than RS cells, FS cells concomitantly have little jitter in the late stage of a transient input (cf. Fig. 10, A and B). Thus threshold dynamics has far-reaching consequences on spike-time variability in RS and FS cells.

Simplified computational models of neural dynamics support this scenario (Ermentrout 1996; Gutkin and Ermentrout 1998; Gutkin et al. 2003; Robinson 2004; Robinson and Harsch 2002). For example, type 1 and type 2 Morris–Lecar (ML) models (Morris and Lecar 1981) capture the mathematical essence of thresholds (bifurcations) of regular firing, using just two variables, one representing voltage and Ca²⁺ channel activation and the other representing K⁺ channel activation. A small change in model parameters causes a shift from type 1 to type 2 behavior and profoundly changes the pattern of dispersion of the final spike times. This result can demonstrate how type 2 neurons intrinsically prefer to stay coherent or be silent, whereas type 1 neurons have a smooth transition between the two extremes. In addition, for the type 2 ML model, the dispersion of the final spikes is very robust against changes in the shape of the stimulus waveform, whereas the dispersion of spike times for the type 1 model is highly sensitive to the exact shape of the stimulus (data not shown). Thus the type of threshold dynamics should have a major impact on the reliability of spike generation and spike timing in the cortex. In the in vivo situation, neurons can receive a greater bombardment by synaptic input and are subject to control by neuromodulators, which could profoundly alter their discharge properties. Nevertheless, under controlled in vitro conditions, the threshold dynamical type of RS and FS cells is a crucial feature of their distinctive patterns of reliability in responses to natural-like input.

In conclusion, in this study, we have compared the variability and reliability of responses of RS and FS cells to complex, natural-like stimuli. Both cell types showed the same pattern of increased CV of interspike intervals in response to higher input synchrony, and this effect is thus independent of the threshold dynamical type. However, there were striking differences between the two cell types in the precision and reliability of spiking during the processing of complex inputs, which may be understood as a consequence of the type 2 property of subthreshold oscillations in FS cells. In FS cells, we found that spike-time jitter is kept low even for late spikes during transient inputs, as expected for type 2 models. In FS cells, but not RS cells, an optimum was found in the level of inhibition required to achieve maximum precision, presumably a result of tuning membrane properties to maximize the amplitude of subthreshold oscillations. Finally, in contrast to RS neurons (de Polavieja et al. 2005), spike shape in FS neurons was hardly modified by the level of excitatory or inhibitory input conductance, meaning that significant spike shape encoding cannot occur. These differences between the two cell types are consistent with a role of RS neurons as input encoding integrators and a role of FS neurons as resonators controlling the coherence of synchronous firing.

	GRANTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
This work was supported in part by the Japanese Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Exploratory Research 17650085/2005, and by European Community Grant FP6.

	ACKNOWLEDGMENTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

	FOOTNOTES

 
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

Address for reprint requests and other correspondence: T. Tateno, Department of Mechanical Science and Bioengineering, Graduate School of Engineering Science, Osaka University, Osaka, Japan, 1–3, Machikaneyama-cho, Toyonaka-shi, 560–8531 Japan

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