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Institute of Physiology, University of Bern, Bühlplatz 5, Switzerland
Submitted 30 April 2006; accepted in final form 16 June 2006
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
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and references therein), these processes have been found more recently in cat auditory cortex neurons, where stimulus-specific adaptation accounts for the difference in responses to rare versus common stimuli over an extended range of time scales (a few hundred milliseconds to hundreds of seconds) (Ulanovsky et al. 2004). In cat auditory nerve fibers (Lowen and Teich 1996) and in the lateral geniculate nucleus (LGN) of the cat thalamus (Lowen et al. 2001), neural activity can exhibit long-range correlations spanning multiple time scales.
Having been collected in vivo, this experimental evidence raises the possibility that the interaction between neurons could underlie the expression of multiple time scales. However, the existence of ion channels with different kinetics suggests that individual neurons may be equipped to produce dynamic activity which correlates on multiple time scales. For example, Spain et al. (1991) reported the existence of (at least) two transient potassium currents in the large pyramidal (PYR) neurons of the layer 5 of cat sensorimotor cortex. These currents decayed with different time scales (
20 ms vs. 10 s) comparable to the time constants of the fastest and slowest adaptation processes previously found in PYR neurons by Rauch et al. (2003). However, no quantitative accounts of the number and the properties of those processes have been reported nor has a similar study been undertaken in neurons the firing patterns of which distinctively differ from the pyramidal or the sensory type, like e.g., fast spiking (FS) interneurons (McCormick et al. 1985).
We analyzed in detail the response of FS and PYR neurons in vitro to long-lasting, noisy stimuli with stationary statistics. The noisy stimuli are meant to imitate the synaptic currents that are observed in vivo in intracellular recordings (Destexhe et al. 2001; Paré et al. 1998). When the input currents were strong enough, the neural response was highly nonstationary for both FS and PYR neurons. The largest variations of the firing frequency over time were observed in the initial phase of the stimulation, but detectable nonstationarities could be seen in the late response of the stimulated cell. This is an indication that several mechanisms operating on multiple time scales determine the response of a neural cell to a stimulus with stationary statistics. We used a novel approach to characterize quantitatively the neuronal response. It consists of two steps: first, a simple model of an integrate-and-fire (IF) neuron was fitted to the late, quasi-stationary responses of the neurons to a variety of input currents (Rauch et al. 2003). Dynamic components were ignored because the variations of the firing rate, although still detectable, were small compared with the firing rates themselves. The second step is an extended model with time-varying processes (adaptation) that was fitted to the temporal response of the neurons from the same set of data, with the neuron parameters set by the previous fit. This second step gave the parameters of the time-varying processes (magnitude and time constant). Based on a detailed analysis performed on those parameters, we report quantitative evidence that multiple time scales are at play in both PYR and FS neurons. We also show that a simple spiking model can provide quite a detailed account of firing rate dynamics of cortical neurons probed in an in-vivo-like environment. Finally, we study the distribution of the functional parameters across cells.
Multiple time scales in cortical neurons might profoundly affect sensory processing and, more generally, neural computation, as proposed by recent theoretical studies (e.g., Brenner et al. 2000; Drew and Abbott 2006; Fairhall et al. 2001) that we briefly review in the Discussion.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
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The experimental preparation was as described previously (Rauch et al. 2003). Briefly, 300 µm thick parasagittal slices of rat somatosensory cortex were prepared from 15- to 40-day-old female and male Wistar rats according to the institutional guidelines. The preparation was done in ice cold extracellular solution using a Campden vibratome (752M, Campden Instruments). Slices were incubated at 37° for 25 min and afterward left at room temperature until transferred to the recording chamber. The cells were visualized by infrared differential interference contrast videomicroscopy using a Newvicon camera (C2400, Hamamatsu City, Japan) and an infrared filter (RG9, Schott Mainz) mounted on an upright microscope (Axioscope FS, Zeiss).
We recorded in current-clamp whole cell configuration from the soma of layer 5 (L5) and layer 2/3 (L2/3) FS (McCormick et al. 1985) neurons and L5 PYR neurons. Recordings and stimulations were made with an Axoclamp-2A amplifier (Axon Instruments) in combination with Clampex 8 (Axon Instruments). The access resistance and the capacitance were compensated using the bridge balance and the capacitance neutralization after having established the whole cell configuration. The data were low-pass filtered at 2.5 kHz with sampling frequency twice the filter frequency. The temperature of the external solution was 31–33° C. Neurons were visually identified, most of them were filled with biocytin (10 mM) and then stained according to the ABC procedure (Hsu et al. 1981). A standard criterion for classification of cells as FS neurons was used (see, e.g., Descalzo et al. 2005): short duration of action potentials (<0.5 ms at half height), fast after-spike repolarization, absence of firing rate adaptation in the first milliseconds of the spike train, a steep frequency-current curve around rheobase (see Figs. 1 and 4).
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Stimulation protocol and main observables
The stimuli were fluctuating in time around an average value but were stationary in the following sense: they were completely characterized by their means and SDs, which were held fixed throughout the stimulation interval. We chose stimuli with virtually no temporal correlation, i.e., resembling white noise. Specifically, the input current was generated as an Ornstein-Uhlenbeck stochastic process by iterating the following expression
 | (1) |
t is a unitary Gauss distributed random variable, updated at every time step. The process was generated and injected at a rate of 5 kHz (
t = 0.2 ms) and the correlation length 
I was 1 ms. The resulting current I(t) has a stationary Gauss distribution with mean mI and variance sI2 (Cox and Miller 1965). Such a noisy stimulus is meant to emulate the synaptic barrage targeting a cortical neuron in vivo due to fast excitatory and inhibitory inputs (Rauch et al. 2003). In a few cells we used I = 5 ms with similar results.
The {mI,sI} space was systematically explored as follows: data points were collected at fixed sI (ranging from 0 to 200 pA), stepwise increasing mI from a subthreshold value up to nonstationary frequencies. The data with sI = 0 (step current) were used to determine the threshold mean current Irh (the rheobase current), and the whole procedure served to establish the whole space of suitable {mI,sI} pairs for the neuron under study. The pool of suitable {mI,sI} pairs was then discretized and explored in random order to prevent correlations between time and one of the two parameters mI,sI. This randomized protocol was used to characterize the response function of the neuron, i.e., its mean firing rate as a function of mI and sI (more details in the following text). For each pair {mI,sI}, stimulation lasted 4 s. Between recordings, the stimulus was switched off and the neuron let to rest for 50–60 s. Some of the recordings were longer (
10 s) to check that the high spike frequencies found could be sustained for longer stimulations. The first part of the neuronal response (0.5 s) was discarded when estimating the mean spike frequency.
The mean spike frequency, f, was estimated as the ratio between the total number of action potentials Nsp and the stimulus duration T. The confidence intervals (68%) of the experimentally measured frequencies were approximately given by = (f+ + f–)/2 with (Rauch et al. 2003)
 | (2) |
Nsp), the longer the observation interval T, the smaller the confidence intervals; second, if no action potentials are observed in (0,T), then the "true" output rate falls within the interval (0,1/T) with 68% confidence. A derivation of Eq. 2 is given in the APPENDIX. Particular care was taken to ensure that the response of the cell was consistent throughout the whole recording session. The cells were classified as consistent if: less than half of the pairwise differences between repeated recordings were out of the error range given by Eq. 2; at a given mI, responses to current with different sI preserved the sign, i.e., if f(mI,sI) > f(mI,sI') in one recording, then the same order relationship is observed in all of the repeats of the same recording [f(mI,sI) is the mean firing rate in response to a current characterized by mI and sI)]. Only consistent cells were further analyzed.
A neuron's response function is the collection of quasi-stationary firing rates in response to a set of stimuli with different means and SDs. We defined a spike train as quasi-stationary if its firing rate was constant over the recording interval or, in the presence of adaptation, if its maximal decrease in firing rate per second, 
f, was below a given threshold, chosen to be 1 Hz/s for PYR neurons and 5 Hz/s for FS neurons (see also following text). The amount of frequency reduction was quantified by the index f = (finit – ffinal)/T, where T = tfinal – tinit, and finit/final is the frequency in a temporal window 1 s wide, centered around tinit/final (for example, if T = 4 s was the total duration of the stimulus, tinit = 1 s, tfinal = 3.5 s, T = 2.5 s). Although firing rate changes could be detected when the slowest components of adaptation were analyzed, they did not affect significantly the quality of the model fits to the response functions (the model fits are described in the next subsections). This was also the criterion for choosing the aforementioned thresholds for f. Pearson's linear correlation and nonparametric Kendall's tau were used to assess the correlation between parameter f and output firing rate f (across all recordings of each single neuron).
The coefficient of variability (CV) of the interspike intervals (ISIs) was estimated as the ratio between the SD and the mean of the ISIs. An initial portion of the spike train of 0.5 s was removed to evaluate the CV on a stationary or quasi-stationary spike train only. The interval was chosen to be significantly longer than the duration of the fastest adaptation process (see RESULTS for more details).
Model response function
A leaky IF (LIF) model neuron endowed with a phenomenological model of spike frequency adaptation (Ermentrout 1998; Fuhrmann et al. 2002; Rauch et al. 2003) was used to describe quantitatively the response of the neurons. The output rate of the LIF neuron in quasi-stationary conditions, f, is described accurately by the solution of the self-consistent equation (La Camera et al. 2004a; Rauch et al. 2003)
 | (3) |
(pA · s) is a parameter quantifying adaptation, and 
is the response function of the LIF model neuron, i.e.,
 | (4) |
I = sI 2I. The meaning of the parameters is as follows: C is the membrane capacitance, I is the time constant of the input current (Eq. 1), r is the absolute refractory period, 
is the threshold for spike emission, Vr is the reset potential after spike emission, is the membrane time constant, and erf(x) = 
is the error function. Data analysis and fitting procedure
The theoretical f(mI,sI) curves were fitted to the quasi-stationary data of consistent cells. The fit was achieved through a Montecarlo minimization (see e.g., Press et al. 1992) of the square difference between the measured (fkexp) and the model (fkth) firing rate
 | (5) |
= {r, Vr, C, , }. k = (fk+ + fk–)/2 amounts to half the confidence interval for point k, with fk± given by Eq. 2. Because only five of the six parameters of the model neurons are independent (the response function Eq. 4 is invariant under the scaling 
, Vr Vr, C C/ with > 0), we set the threshold to = 20 mV without loss of generality.
The minimum 
min2 of the quantity (5) with respect to the parameters set 
follows approximately a 2 distribution with N-M degrees of freedom, where N is the number of experimental points, and M = 5 is the number of free parameters. The fit was accepted whenever the probability P(N-M2 
min2) was >0.01. An unequal variance t-test was used to detect differences in average neuron parameters (the same results were obtained with an equal variance t-test). Significance was taken at 5% level.
The input resistance of the neurons was calculated from the voltage transients in response to at least six different hyperpolarizing current pulses (600-ms duration, average of the last 300 ms; step amplitude: 20 pA). The membrane time constant was estimated by injecting brief (0.4 ms) hyperpolarizing current pulses (–2.5 nA) into the soma. From the averaged (n = 50) decaying voltage transient after this current pulse, was obtained from the slope of a straight line fitted to the tail portion of the semilogarithmic plot of the membrane voltage against time (Iansek and Redman 1973). The membrane capacitance was then obtained as the ratio between the membrane time constant and the input resistance. We refer to these values as to the directly estimated parameters in the following, as opposed to the effective parameters obtained from the fit of the model to the response functions.
Full model fit of the temporal response
The model neuron having the quasi-stationary firing rate given by Eqs. 3 and 4 is described by a single variable, the membrane voltage V, which below the spike threshold obeys
 | (6) |
I = sI2I (symbols have the same meaning as in Eqs. 1 and 4). I is a feedback current driven by the neuron's instantaneous output rate
 | (7) |
can be interpreted as being (proportional to) the internal concentration of the ion species responsible for adaptation (see e.g., Powers et al. 1999; Sanchez-Vives et al. 2000; Sawczuk et al. 1997; Schwindt et al. 1989); 
as its peak conductance, and as its decay time constant to zero between action potentials. When V = , a spike is said to be emitted, and V is reset to Vr, where it is clamped for r ms. The output rate reaches eventually the quasi-stationary value given by Eq. 3 and 4 with = (for a comprehensive discussion of this model and its validation in the presence of noise, see La Camera et al. 2004a).
This model, with the neuron parameters as determined by the fit of the quasi-stationary response function (see the previous section), was used to fit the instantaneous firing rate, defined as the inverse of the ISI as a function of time, f(t) = 1/ISI(t). For illustration purposes only, a smoothed version of f(t), obtained by using a Savitzky-Golay polynomial smoothing filter (see e.g., Press et al. 1992) was used in Figs. 1–10. Note that the theoretical ISIs depend on the model neuron parameters = {r, Vr, C, , } (see Data analysis and fitting procedure). Including these as free parameters in the fitting procedure could have resulted in an over-fitting of the temporal properties. Moreover, neuron parameters should be the same, both after the response has reached its quasi-stationary characteristics, and during its temporal evolution. For this reasons, previously fitted neural parameters were held constant during the analysis of the temporal response. This limited the analysis to only those neurons with stable and fitted quasi-stationary response functions. Some of the PYR neurons studied by Rauch et al. (2003) could not be fitted according to the 2 test but were well described by the model in terms of average absolute difference between experimental and theoretical firing rates (<1.5Hz per data point if the frequencies were <50 Hz, <2.5 Hz per data point for all the other frequencies). These neurons were included in the present analysis, giving a total number of nr = 92 single recordings from n = 21 PYR neurons.
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in Eq. 6 was the sum of independent components, 
, each of which obeying an equation like Eq. 7, with corresponding k, k such that kk ''' k, and 
(negative ks correspond to facilitating processes). This constraint ensures that the quasi-stationary firing rate of each spike train (reached when t >> maxkk) agrees with that given by Eq. 3. Note that it is necessary to consider the right number of processes to estimate the time constants involved because 
, and this constraint will be satisfied with different values of k depending on n, the total number of processes. The relative percentage of a process was defined as k,% ''' 100|k|/
Unlike the neuron parameters, which were obtained from the fit to the quasi-stationary response and were therefore independent of firing rate, the parameters characterizing the temporal response were fitted separately for each recording and therefore were, at least in principle, firing-rate and/or input-dependent. Their properties as functions of firing rate and input current are analyzed in detail in RESULTS.
The fit of the temporal response was achieved via a Montecarlo procedure as described earlier (see Data analysis and fitting procedure), using as objective function the squared difference of the ISIs, i.e., 
. ISIth(t) was obtained by simulating the full IF model described in this section. The quality of the fit was assessed by eye and by comparison of the best-fit value of ISI2 with the number of degrees of freedom N-M (N is the number of ISIs in a given spike train and M is the number of parameters to be tuned, described in the following text). Most fits of the responses to step-like stimulation gave values of 2 around or less than N-M. This would make the fits acceptable if the errors on the single experimental ISIs, not available, would be around unity (in ms) (Press et al. 1992): a conservative value because it is comparable to the duration of an action potential. The (unknown) error in the ISIs would most likely be higher and hence the observed 2 smaller, making the test more significant. The exact reproduction of spike times, as assessed, e.g., through the Victor metric and similar spike train distances (see Victor 2005 for a review), was not the aim of the fitting procedure. Parameters to be tuned were the ks (max n = 3), the ks (n – 1, the nth being determined by 
, plus one additional parameter, mI, representing a constant offset current. The input current to the model neuron was I + mI, where I was exactly the current injected during the experiment (i.e., the term mI · dt/C must be added to the right hand side of Eq. 6). [mI turned out to be different from zero only in PYR neurons to compensate for the strong burst-like initial adaptation which involves only the first one to three ISIs (most commonly a doublet of spikes, see RESULTS) (see also Rauch et al. 2003), not captured by model Eq. 7. This subtraction procedure compensates for the initial adaptation current in its steady state while neglecting the initial transient during the first few action potentials. No corrective term was required for fast spikers, consistent with the absence of strong initial adaptation in these neurons.
The peak conductances k were given by the ratio k/k (negative if k < 0, representing facilitation instead of adaptation). In total, this makes M = 2n parameters to be fitted. For n = 3 processes (the maximum number of processes found to be required), M = 6. The number of ISIs (N) was in the range 300–600 for PYR neurons (30–60 Hz for 10 s); 400–800 (4 s) or 1,000–2,000 (10 s) for FS neurons (100–200 Hz). For each cell, four to eight different spike trains were analyzed, often with repetitions of the same input to compare intra-cell with inter-cell variability. The repetitions were never consecutive; FS neurons had typically a consistent response to the same stimulus, whereas PYR neurons would often display different time constants and amounts of the processes involved, although within the same order of magnitude. Most fits were obtained in case of step current stimulation, and then the best-fit parameters were used as the initial guess to fit the responses to noisy current.
Dependence of the adaptation/facilitation parameters on input current
Linear correlation of the adaptation/facilitation parameters |i| and i (indicated generically with the symbol Y in the following), with output firing rate and input current (across all recordings of all neurons of the same type, i.e., either FS or PYR) was taken as a measure of the dependence of those parameters on output firing rate and input current, respectively. Because this analysis involved multiple comparisons, significance was taken at 0.5% level. The details of the analysis are reported in the APPENDIX. The dependence of Y on input current, x ''' mI – Irh, was then summarized either in the form of a linear relationship, Y(x) = ax + b (if Y was positively correlated with x), or in the form of an exponential function, Y(x) = ae–bx (if anticorrelated with x), where a,b are constants. By interpolation, this provides the function Y(I+) for all input currents in the physiological range, where I+ ''' I – Irh. Parameters a and b were obtained through a standard least-square procedure. The functions Y(x) were chosen because of the following observations: first, adaptation/facilitation can be observed only when the neuron is firing, hence it is more appropriate to describe them as functions of mI – Irh as opposed to mI; moreover, this avoids the problem of possibly having negative values of Y above rheobase, i.e., for mI > Irh (the parameters are positive by definition); second, increasing variables were, in the domain of interest, well described by their regression lines; and third, exponential curves provided fits almost as good as the regression lines to decreasing variables at the same time preventing them from reaching negative values at large inputs.
In the absence of correlation between Y and x = mI – Irh, we summarized the statistical properties of Y by reporting the (combination of) truncated Gaussian distributions that approximate the experimental distribution. The Gaussian distributions were truncated at negative values because the parameters are positive by definition. Although it would be more appropriate to use a positive-definite distribution like, e.g., the Gamma distribution, in our data, there was little practical difference between the Gamma and the truncated Gaussian distribution that approximates it, and for the sake of simplicity we chose the latter.
The experimental distributions shown in Fig. 8 were obtained by smoothing the data with a Gaussian filter at half the optimal bandwidth, to reveal features such as multiple modes (Bowman and Azzalini 1997). For the red curve in Fig. 8, bottom right, the optimal bandwidth was used. Correlations and other statistical analyses were performed on the raw data.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
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The response of n = 15 cells from the L5 of rat somatosensory cortex (P28-43) were collected using the random step protocol with fluctuating current described in METHODS. n = 10/15 were identified as FS cells, showed a consistent response function (see METHODS), and were further analyzed. A standard criterion for classification of cells as FS neurons was used (see e.g., Descalzo et al. 2005): short-duration of action potentials (<0.5 ms), fast after-spike repolarization, absence of firing rate adaptation in the first milliseconds of the spike train, a steep frequency-current curve around rheobase (Figs. 1 and 4). We recorded also from n = 12 neurons in L2/3 (P16-27), 9 of which were clearly identified as FS cells as described in the preceding text. n = 5/9 neurons showed a consistent response function and were considered for further analysis. L5 and L2/3 interneurons had similar physiological parameters (i.e., directly estimated membrane time constant and capacitance, see METHODS and Table 1, last 3 rows), were injected with currents of similar magnitude and SD, and showed similar response patterns. As a consequence, their properties were summarized together unless explicitly noted.
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4 s) step- and noisy-current injections are shown respectively in Figs. 1 and 2. FS interneurons showed no initial adaptation (defined as a strong but transient decrease in firing rate that involves the first few spikes only) in accord with what reported in previous studies (e.g., McCormick et al. 1985). Nevertheless, the cells showed consistent slow reduction in firing rate over time for stimuli longer than a few hundred milliseconds, see Fig. 1, A and B (experiment). This behavior is well captured by the simple adapting model described in METHODS (Eqs. 6 and 7; Fig. 1, model) with an adaptation time constant = 2.2 s. The presence of slow adaptation in FS neurons has been reported only recently by Descalzo et al. (2005) in ferret visual cortex, both in vitro and in vivo; preliminary evidence in rat somatosensory cortex in vitro had been reported by Reutimann et al. (2004).
In a fraction (n = 5/15) of interneurons one adapting process was not enough to capture the time course of the firing rate, and the generalized model described in METHODS had to be used. Two components were sufficient to reproduce the data, one with a time constant of 100–300 ms, the other slower and spanning a large range of time constants ( 0.8–30 s). Nine cells of 15 showed stuttering behavior (see Fig. 5, left, inset), in response to low noise current (sI = 0–30 pA). Such behavior was consistently suppressed in the presence of higher input fluctuations (sI = 50–200 pA) as found for PYR neurons (Rauch et al. 2003). A more detailed analysis of the temporal properties of the response will be reported in later sections.
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f = (finit – ffinal)/T (see METHODS), which provided a measure of late (or slow) adaptation. For the sake of subsequent analysis, nonstationary responses were considered as quasi-stationary if their inclusion in the fitting procedure described in Data analysis and fitting procedure did not alter the outcome of the fit (see also the next section). Typically, in FS interneurons this happened for f smaller than 5 Hz/s (10 Hz/s in a few L2/3 interneurons) at maximal frequency (150–200 Hz); in PYR cells, for f = 1 Hz/s at maximal frequency (40–60 Hz). Hence the decrease in firing rate over time after the initial transient was much more pronounced in FS cells than in PYR neurons (see also Fig. 6). In n = 10/15 interneurons, f values were strongly correlated with the quasi-stationary output rate f; in eight of these neurons, correlations were strong but significant only at low values of sI (Pearson 
and Kendall's tau, P < 0.05). Note that although a large amount of noise typically weakened the correlation between f and f, it did not decrease the amount of maximal adaptation (quantified by the maximal f).
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The response function of the modified LIF neuron was fitted to the response of consistent cells as described in METHODS. A high percentage of cells could be fitted in both L5 and L2/3. Results are summarized in Table 1, where also the effective parameters for PYR neurons taken from Rauch et al. (2003) are reported. A few examples are shown in Fig. 3. L5 and L2/3 FS neurons differ significantly in the (effective) capacitance C (smaller in L5), the membrane resistance R (larger in L5), and in the rheobase current Irh (smaller in L5). These differences could in part be accounted for by the different age of the rats in the two groups. It is interesting to note that although the directly estimated C, , R had similar values for L5 and L2/3 FS neurons, the effective C, R were the only parameters which were significantly different between L5 and L2/3 FS cells (t-test, P < 0.05). A discrepancy between directly estimated and effective parameters was found also in PYR neurons (Rauch et al. 2003).
Comparison between the response functions of FS and PYR neurons
The "average" response functions of PYR and FS neurons are shown in Fig. 4, where the average best-fit parameters from Table 1 were used. The maximal amount of noise which could be used for FS interneurons was sI = 200 pA, as opposed to 500 for pyramids. However, the response at rheobase for the maximal sI used (50 Hz) was stronger than for PYR neurons (10 Hz). This can be attributed to a significantly smaller membrane time constant in FS neurons (Table 1; t-test, P < 0.002), mostly due to a smaller membrane capacitance. Moreover, because of a larger input resistance, FS neurons have a smaller rheobase current. Taken together, these factors imply that FS neurons respond faster and to a much higher extent to any kind of change in the input than PYR neurons. Consistently, FS cells had smaller effective C, , and r, than PYR neurons (Table 1).
Note that is smaller in FS neurons, despite the larger index f quantifying slow adaptation (1 Hz/s for pyramids vs. 5–10 Hz/s for fast spikers). This is the consequence of the steeper slope of FS cells' response function (Fig. 4). Indeed, to quantify adaptation, both and the slope of the response function are required. According to Eq. 3, where the non-adapted response function appears, the adapted firing rate f is given by the nonadapted response at mI' = mI – f, so that a steeper response function requires a smaller value of to produce the same amount of adaptation. In Fig. 3, one can notice that the values are indeed anticorrelated with the slopes of the corresponding response functions. The results of this section point to a faster and larger response of FS neurons to in-vivo-like current, compared with PYR neurons.
Variability of the ISIs
We also measured the ISI variability of FS neurons in response to in-vivo-like current, through its coefficient of variability (CV, see METHODS). Strictly speaking, this measure can be meaningfully applied only to stationary spike trains (Gabbiani and Koch 1998). Given the presence of several temporal processes, the CV was calculated on the portion of spike train left after the removal of the first 500 ms, an initial transient of the duration of the faster adaptation/facilitation processes (see the next section and Table 2 for details). This allows us to deal with the variations due to slow adaptation of the firing rate only. Two typical cases are shown in Fig. 5 for one L5 (left) and one L2/3 (right) FS interneuron, together with the prediction of the LIF model neuron the parameters of which were tuned to fit the firing rates only (full lines). Only one adapting current was used in the model Eqs. 6 and 7. The use of more adapting/facilitating components to fit the temporal response did not significantly improve the match to the CV (not shown). In the model with one adaptation component, the CV slightly increases with up to 500 ms, which was the value used in Fig. 5 to get the best match with the data. This type of behavior is expected for larger than the membrane time constant (Liu and Wang 2001). Above = 500 ms, no change in variability was observed.
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; Gur et al. 1997; Holt et al. 1996; Lee et al. 1998; Shinimoto et al. 2003, 2005; Wiener et al. 2001). For larger output rates (200 Hz), the coefficient of variability drops to values that are still >0.5, if there is enough variability in the input. High ISI variability in the intact brain is believed to be the result of a balanced synaptic input current, characterized by large fluctuations around a sub-threshold average (e.g., Amit and Brunel 1997; Destexhe et al. 2001; Haider et al. 2006; Holt et al. 1996; Lerchner et al. 2006; Shadlen and Newsome 1994, 1998; Shu et al. 2003; van Vreeswijk and Sompolinsky 1996). Comparison between the cases of small and large input fluctuations in the data of Fig. 5 supports this view.
It is also interesting to compare this result with the typical variability of a L5 PYR neuron (dashed lines in left panel). The maximal CV is <0.5 even though s = 300 pA was used (for s = 400 pA, CV 0.6, but only for f < 30 Hz) (see Rauch et al. 2003). As opposed, for the L5 FS cell in the left panel, CV 0.8 for f 80 Hz, the maximal stationary firing rate found for a L5 PYR neuron. These results are consistent with the higher sensitivity to fluctuations found in FS neurons compared with PYR neurons. Interneurons in L2/3 tend to be less variable than in L5, especially around maximal frequency (typically CV 0.5 compared with 0.6 for inputs with maximal sI). A few spike trains in response to low-noise input showed high variability due to stuttering behavior (Fig. 5, inset), that cannot be reproduced by the model.
Dynamics of the response: adaptation/facilitation of the firing rate
So far we have been concerned with the properties of the quasi-stationary response as reached after prolonged exposure to a stationary stimulus, and only mentioned in passing the temporal dynamics through which the quasi-stationary behavior was reached. In this section, we characterize the dynamic processes involved, summarizing some of their statistical properties.
PYR and FS neurons showed different temporal patterns in response to a sustained input with stationary statistics. A comparison is shown in Fig. 6. Note the phasic response of the PYR neuron, characterized by an extremely fast and strong initial adaptation, followed by an increase in firing rate, and then by a slow decay. FS interneurons lacked the strong initial phase, and were characterized by tonic firing, although of slowly decreasing rate over time. To capture these behaviors, we extended the model IF neuron to include a minimal number np of independent processes, sufficient to characterize the temporal response. In order for the extended model to have the same quasi-stationary firing rate as the original model used to fit the response functions of Fig. 3, the same neuron parameters were used, including 
(see Full model fit of the temporal response). This left np – 1 independent i and their time constants to be adjusted by fitting the instantaneous firing rate. Two examples of this fitting procedure are shown in Fig. 7, for a noisy-driven L5 FS cell (top) and a step-driven PYR neuron (bottom). A summary of the results is reported in Table 2.
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50–200 ms), early facilitation (F, 0.5–1 s), and slow (or late) adaptation (sA, order of seconds). The adaptation/facilitation components became clearly visible above a critical firing rate, 30 Hz for PYR neurons (all processes) and above 40 Hz (sA) or 80 Hz (fA) for FS interneurons (with no difference between L5 and L2/3). We next discuss their properties in turn.1
INITIAL ADAPTATION.
Initial adaptation (iA), defined as a burst-like adapting process affecting only the first one to three ISIs (Rauch et al. 2003; Sawczuk et al. 1997), was present in 13/21 PYR neurons [nr = 15 recordings, across which initial frequency was fi > 52 ± 24 (SD) Hz, with a minimal value of fi = 17 Hz], and absent in FS neurons. The model Eq. 7 is inadequate to describe iA, therefore we used a constant offset current mI, which activates immediately after the onset of the stimulus (see Full model fit of the temporal response).
FAST ADAPTATION.
Fast adaptation (fA). Fast- and slow-duration adaptation mechanisms are well documented in the literature on PYR neurons but only recently were reported in FS neurons (Descalzo et al. 2005; Reutimann et al. 2004). The longest time constants of fA were 132 ms (PYR) and 260 ms (FS), whereas the shortest time constant of sA was 1 s in all cells. The two distributions were clearly separated around the value of 0.5 s, which was then used to discriminate fA from sA. Fast adaptation was slower in FS neurons compared with pyramids, fA 200 (FS) versus 50 ms (PYR), see Table 2; was always present in PYR neurons, while present only in 30% of interneurons (for output rates larger than 80 Hz), where it contributed 30% of the total adaptation (vs. 50% in PYR neurons, see Table 2).
SLOW ADAPTATION.
Slow adaptation (sA) was present in all neurons, for output rates larger than 30 Hz (PYR) or 50 Hz (FS). In PYR neurons, sA spanned a unimodal range of time constants, 1.4–15.7 s (n = 17/17 cells, nr = 66 recordings), peaked around 6 s (see Fig. 8). In FS interneurons, sA spanned a bimodal range of time constants (mostly in the range 0.8–15 s, with 30 s in a few cases; n = 14/14, nr = 60), peaked around sA(1) 2 and sA(2) 12 s, respectively (see Fig. 8 and Table 2). This suggests that sA in FS neurons could be divided further into two processes. We will take this possibility into account in a later section, when attempting a model of the dependence of sA on input current.
EARLY FACILITATION.
Early facilitation (F) manifests itself with a progressive shortening of the ISIs during, approximately, the first second of stimulation, after which the ISIs either remain constant or start to increase steadily (signature of sA), sometimes until a new plateau has been reached (quasi-stationary points). Facilitation was found in all PYR neurons, but in none of the FS neurons. The time constant of the process ranged from a few hundreds milliseconds to 1 s (n = 17/17, nr = 77), and accounted for 37% of the adaptation/facilitation processes, with an extended range (12–100%; see Table 2). Purely facilitating spike trains were observed at low frequencies, 20–30 Hz, where neither iA nor sA made an observable contribution (n = 2 neurons). Facilitation was found also at high initial rates (120 Hz), where both iA and sA were strongly present. All three processes were present if the output rate was above a critical value 30 Hz, as in Fig. 7, bottom (occasionally, also at lower output rates). In such cases, the spike train toward the end of stimulation was typically adapted, evidencing a larger effect of adaptation versus facilitation on the final output rate. F was present also when other pipette solutions were used, i.e., in the presence of 135 mM of K-methylsulfate, or in the presence of 10 mM EGTA, with similar time constants (Rauch et al. 2003). In both PYR and FS neurons, there was occasionally evidence of a delayed response of the first action potential after stimulus onset, especially for stimulus strength around rheobase. Such a behavior is to be kept distinct from facilitation, given the absence of (transient) monotonic decrease of ISI duration after the onset of the delayed response.
A few outlier cells had atypical behavior and were not included in the analysis of Table 2. Instead of sA, two PYR neurons had slow facilitation ( > 20 s), whereas a third neuron had two components of faster adaptation (nr = 4 recordings); moreover, facilitation was rare in this neuron, and visible only at an unusually large initial firing rate of 100 Hz. One L2/3 FS cell showed intermediate-duration adaptation of 500 ms.
Interaction between the adaptation/facilitation processes
For both FS and PYR neurons, the percentages of the three processes (fA, F, and sA) were mutually (i.e., pair-wise) anticorrelated (for example, for PYR neurons –0.4, P 10–3): the more of one process, the less of the others. In PYR neurons, the higher the output rate, the more (and faster) sA, the less (and slower) F, whereas fA did not depend on firing rate (see the APPENDIX for details). At high firing rates, sustained activity was overtly affected by slow adaptation. As opposed to PYR neurons, no clear relation to firing rate emerged in FS neurons. The %s were not correlated with output rate, and no pairwise correlations were found among the sA parameters, whether the time constants were considered separately (sA(1) and sA(2)) or not (see the previous section). Recordings with larger output rates had a tendency to be dominated by sA, with less (but faster, as opposed to PYR neurons) fA.
Effect of input fluctuations on adaptation
In FS neurons, neither fA nor sA were correlated with average input current (either mI or mI – Irh; details are deferred to the next section). However, fast adaptation was found only in the presence of small fluctuations (sI < 60 pA), whereas slower adaptation was just as strong irrespective of input fluctuations. This suggests that the input fluctuations play a role in selecting the type of adaptation taking place, independently of average input current (for each sI, the same spectrum of mI values were used). However, there was no clear relationship between the amount of fA and the value of sI (not shown).
We found no evidence of a similar effect in PYR neurons. In the presence of large input fluctuations, the clear phasic response of Figs. 6 and 7 is more difficult to make out, but initial adaptation was still perceivable (data not shown). A comparison of the duration of the first few ISIs compared with the subsequent few ISIs, in the presence and in the absence of fluctuations, showed no appreciable difference in the two conditions, pointing to the presence of the same temporal processes in both cases. Specifically, we counted the number n1 of ISIs in the first 100 ms (the order of magnitude of duration of fA), calculated their cumulative duration T1, and compared T1 with the cumulative duration T2 of the same number n1 of subsequent ISIs. A spike train was considered fast adapting if T1 < 0.75 T2 (a conservative measure, aimed at preventing taking into account spikes due to the occasional excess of upward deflections of the input current in the first 100 ms). For each cell, we compared the fraction of fast-adapting spike trains obtained in response to fluctuating and constant stimuli, respectively. These fractions were similar in all of the analyzed neurons and were sometimes larger in case of fluctuating inputs, leading to the conclusion that our data do not support a strong, general effect of input fluctuations on fast adaptation in PYR neurons. Moreover, the slow adaptation index f had its maximal value unaffected by fluctuations, and early facilitation was observed in all recordings. In summary, input fluctuations did not disrupt the cellular properties of adaptation/facilitation in PYR neurons.
Dependence of adaptation/facilitation on input current
In this section, we report how adaptation/facilitation relate to input current. In the following, we indicate with the symbol Y the generic parameter i or i. A dependence of Y on input current was inferred if Y was linearly correlated with mI – Irh (across all recordings of all neurons of the same type, see METHODS). Two examples of correlations are shown in Fig. 9. The details of the correlation analysis are reported in the APPENDIX.
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mI, as explained in Full model fit of the temporal response. mI [–200 ± 210 pA, range (–800, 480) pA] was negatively correlated with mI – Irh (–0.263, P = 0.016), and we chose a linear model for it as there was no need to prevent negative values. The results are summarized in Table 3, where models are reported as Y(x) ± CVY, where Y(x) expresses parameter Y as a function of input current x, and CVY is its coefficient of variability (SD/mean). CVY gives a measure of the variability found across different recordings and different neurons.
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