Rate maintenance and resonance in the entorhinal cortex

Electrophysiological recordings

All experiments were conducted as approved by the UCSD Institutional Animal Care and Use Committee. Young (14- to 21-day-old) Long–Evans rats were anesthetized by overexposure to CO₂ and decapitated. The brain was quickly removed and immersed in cold (0 °C) oxygenated artificial cerebral spinal fluid (ACSF; in mm: NaCl 126; KCl 3; NaH₂PO₄ 1.25; MgSO₄ 2; NaHCO₃ 26; glucose 10; CaCl₂ 2; buffered to pH 7.4 with 95/5% O₂/CO₂). Horizontal slices were prepared using a Vibratome cutter (TPI). Slices were allowed to recover for 1 h prior to recording in a holding chamber at room temperature, continuously bathed in oxygenated ACSF. Slices were transferred to an immersion chamber (RC-27L; Warner Instruments) and visualized with IR-DIC optics (Zeiss Axioskop 2FS Plus, Dage CCD100), maintained at 34 °C (TC-344B; Warner). Electrodes of resistance 4 – 6 MΩ were pulled on a horizontal puller (Sutter Instruments) and filled with a recording solution (in mm: KGluconate 135; KCl 4; NaCl 2; HEPES 10; EGTA 0.2; MgATP 4; GTP-tris 0.3; phosphocreatine-tris 10). Excitatory synaptic transmission was blocked by 6-cyano-2,3-dihydroxy-7-nitroquinoxaline acid (CNQX; 10 μm) and d-2-Amino-5-phosphonovaleric acid (d(-)-APV) (50 μm), produced by Sigma (St Louis, MO, USA).

We obtained whole-cell recordings from superficial entorhinal cortical (EC) layer II neurons, selected by their oblong cell bodies, as well as particular characteristics of their electrophysiological responses to long current steps – a prominent (> 30%) sag in response to both depolarizing and hyperpolarizing current injections (Alonso & Klink, 1993), as well as an early first spike in response to suprathreshold stimuli (Haas & White, 2002). Recordings were made in current-clamp mode. Intracellular signals were amplified (Axoclamp 2B; Molecular Devices), low-pass filtered (eight-pole Butterworth at 5 kHz) and digitized at 10 kHz with a DAQ card (NI PCI-6035E) controlled by lab-made software created in LabView (National Instruments). Series and input resistances, and resting potentials, were monitored throughout each experiment; data from cells with variations > 25% in those parameters were discarded. We compensated for series and access resistance, which was typically 20 – 40 MΩ.

In parallel, we recorded from and stimulated neurons with a real-time Linux-based dynamic clamp (Dorval et al., 2001)_. In dynamic-clamp mode, a current was injected that varied with cellular voltage and time and was recalculated at each acquisition time step (0.1 ms). In this manner it was possible to mimic a synaptic conductance input, rather than command a voltage or current input, to the cell. We delivered excitatory synaptic inputs as double exponential waveforms S(t) governed by an ordinary differential equation of the form , with  = 0.2^ms and  = 5^ms. T represents neurotransmitter in the synapse as a function of V_pre, the presynaptic voltage train (to simplify, we used square pulses rising from −100 to +10 mV for 2 ms at the input times, as presynaptic spikes): (V_t = 0 mV, V_s = 5 mV). The total injected synaptic current was then , with synaptic conductance g_syn = 2 nS, membrane potential V_m and synaptic reversal potential V_syn = 0 mV. Synaptic inputs were added to the current injected through the amplifier; the DC amplitude of the injected current was chosen to elicit steady spiking at ∼5 Hz.

Data were discarded from any cells in which the cell health or recording degraded or failed before having recorded the full set of time-rescaled stimulations for all six inputs (roughly 1 h); this yielded a dataset of six neurons. From the six neurons, the average rest potential was −63.7 ± 4.6 mV, without correction for junction potential. The times of the conductance inputs were taken from six sets of inputs – one regular input with coefficient of variation (standard deviation divided by the mean) C_V = 0, and five different sets of spike times as previously recorded from a cell depolarized with current injection only. To mimic the range of natural variability of cellular spiking, the C_V values of these ‘quasi-regular’ sets covered the range from 0.05 to 0.45, with mean C_V = 0.16. Each train of inputs was delivered repeatedly with temporal scalings chosen such that the input intervals would range between one-third and three times the unperturbed interspike interval (ISI) of the recorded neuron. The baseline current injected to the cell was maintained through all input sets. For a total of six neurons, with six input trains and five to nine (mean 7.19) compressions of each input for each neuron, we delivered a total of 259 inputs, each of 1 min duration. Inputs with different compressions were delivered in random order, to minimize any effects of rate accommodation. Offline analysis was performed in Matlab (Mathworks). Errors are reported as standard deviations (SD) or standard errors of the mean (SEM) where indicated.

ISI-distance

To assess the reliability or similarity between input and output for different input firing rates, we employed a slightly modified version of the ISI-distance, a novel method that has the advantage of being parameter free and time scale independent (Kreuz et al., 2007, 2009).

First, we form a temporal measure of the firing rate of the first spike train . At each moment in time, the ISI is taken as

and likewise for the second spike train .;]> The ISI-ratio between x_ISI and y_ISI is then formed and normalized:

This measure becomes zero at moments when both cells are firing at the same rate, and approaches −1 or 1, respectively, if the firing rate of the first (or second) train is infinitely high and the other infinitely low.

A measure of spike train distance is derived by averaging the value of the ISI-ratio over time:

Note that here, in a variation of the original measure (Kreuz et al., 2007), the absolute value is applied to the whole integral, which yields a measure of frequency mismatch. The original measure is an upper bound to this new variant ;;]> the measures are equal when the ISI-ratios are always positive or always negative (e.g. for periodic spike trains with C_V = 0). This usage also allows us to compute the expected values of ;;]> for periodic spike trains differing only by a constant frequency mismatch, and thus to assess the extent to which differences between input and output cannot be explained as a consequence of pure frequency mismatch.

Spike classification

Spike times were determined as the time when the voltage crossed a fixed threshold value (defined as the minimum value plus half the data range) from below. We classified output spikes into categories of natural, perturbed and forced according to whether they were not preceded, not immediately preceded, or immediately preceded, respectively, by an input. Interestingly, we were able to perform this classification using information from the output voltage trace only (cf. Fig. 4A), a strategy that may be very useful for recordings in which only the output, but not the input, is accessible. To start with, the output voltage trace within each ISI was low-pass filtered by a moving average of 10 time steps (1 ms). Within each ISI, the voltage signal was scanned for a transition from concavity to convexity (i.e. a change of sign from positive to negative of the second derivative), which marked the occurrence of a synaptic input (e.g. at 0.33 s and at 0.61 s in Fig. 4A). If there was no such transition within an ISI (i.e. membrane voltage was concave up for the entire interval), the spike terminating the interval was classified as natural (e.g. the second spike in Fig. 4A). To distinguish between perturbed and forced spikes, the voltage signal between the last transition and the following output spike was examined. In case of a voltage decrease, the following spike was categorized as perturbed (e.g. the third spike in Fig. 4A); otherwise, it was considered to be a forced spike (e.g. the first and the fourth spikes in Fig. 4A). Inputs were subsequently classified as forcing when they preceded a forced spike, perturbing if they were the last input within an ISI but did not trigger an immediate response, or as neutral if they were not the last input within an ISI.

Overall statistics: rate, coefficient of variation and resonance

First, we carefully determined a reasonable dynamic range for our experiments, performed on single cortical neurons in vitro. To parameterize this range, we drove spiking with long injections of varying amplitudes of direct current injection, as a proxy for the activation of internal dynamics and ionic currents underlying regular spiking, in current-clamp recordings. We found that we could easily drive these neurons to fire at 20+ Hz (Fig. 1; further data not shown). The C_V of the ISI distribution was overall inverse to firing rate; for rates of 1–2 Hz, C_V of firing approached 1, while for rates closer to 20 Hz, C_V was ∼0.2.

We chose 5 Hz as a target rate for further experiments for several reasons: first, 5 Hz is within the physiologically and behaviorally relevant theta range of frequencies (Fyhn et al., 2004; Buzsaki, 2005; Hasselmo & Brandon, 2008). Second, it is the lowest rate at which firing is still fairly regular; frequencies below 5 Hz were too irregular to quantify a meaningful value of ISI. At higher frequencies, sensitivity to input is inverse to firing rate (Reyes & Fetz, 1993a). Thus, 5 Hz is a range in which neurons are maximally sensitive to input while firing regularly – enough above spiking threshold to fire, but not so far that sensitivity to input is reduced. To elicit spikes near 5 Hz, we used inputs of 229 ± 28 pA (mean ± SEM, n = 6). The average C_V of unperturbed firing in the cells to which we later delivered conductance inputs was 0.28 ± 0.03 (mean ± SEM, n = 36). Thus, we drove neurons with direct current to a weakly active range, firing near 5 Hz, with plenty of interspike time and dynamic range remaining to respond to synaptic inputs.

Once we stimulated each neuron to fire near 5 Hz, we added strong synaptic inputs via a dynamic clamp on top of the depolarizing current, with input rates varying from one-third to three times the firing rate of the stimulated neuron. These inputs differed from those used in previous studies of reliability in that they were temporally discrete instead of continuous; also, we did not zero-mean them, in order to not constrain neural spike rates (i.e. artificially lower rates for our faster inputs by subtracting DC current). A similar paradigm was used in earlier experiments (Reyes & Fetz, 1993a; Gutkin et al., 2005) to predict modulation of firing times and rates in response to varied sizes of temporally sparse inputs, but these studies sought to avoid entrainment. In contrast, we used only one input strength but varied input rate, seeking to test entrainment specifically. We aimed to test the assumption that strong inputs can more strongly modulate, or ‘hijack’, neuronal firing.

We initially tested several conductances for our synaptic inputs and settled on a final strength of 2 nS, which drove spikes about half of the time. To broadly test the overall efficacy of the synaptic inputs (which will be addressed in a more detailed manner below), we totaled the percentage of spikes preceded by an input within 20 ms and found that this was about 51%. As a null hypothesis test, we computed the percentage of spikes preceded by inputs within 20 ms in shuffled spike trains; this fraction was close to 10%. For the inputs that failed to immediately elicit a spike, the average evoked excitatory postsynaptic potential (EPSP) was 4.9 ± 1.7 mV (mean ± SD, n = 804, measured from baseline to peak). Because these ∼5 mV inputs were delivered to neurons that were already supra-threshold and firing steadily, and often drove spikes directly, we qualify them as strong (Destexhe et al., 2003); our inputs were twice as large as the perturbations used to test phase–response curves, without directly driving spikes, in a similar paradigm (Netoff et al., 2005), and correspond to the largest of inputs used in other similar studies (Reyes & Fetz, 1993b). These results confirm that we provided a strong (but not overwhelming) synaptic input to the spiking neurons.

These choices of firing rate and input size were critical – our intent was to allow inputs to interact with intrinsic dynamics, rather than overpower or overdrive neurons with either the direct current injection or the synaptic conductance inputs.

Subsequently, we delivered six different trains of conductance inputs to each of the six neurons and repeated each of those input trains using different temporal compressions (Fig. 2; see Materials and methods), resulting in 259 input presentations in total. One of these trains was periodic, and the rest had coefficients of variation varying from 0.05 to 0.45, which spans the natural variability in spike times as discussed above.

We plotted the mean output interspike intervals (ISIs) of the driven neurons against the mean input interconductance intervals (ICIs; Fig. 3A). For each stimulus presentation, we normalized both input ICIs and output ISIs to the ISIs of the unperturbed neuron (driven with only direct current injection and no synaptic input; cf. the top trace in Fig. 2). Over the six neurons used, the mean unperturbed ISI was 189 ± 41 ms (mean ± SEM). We found that despite the strong synaptic inputs, neurons generally maintained their natural firing rates for both slower and faster inputs (Fig. 3A). To quantify the effect of input rate on the neuronal firing rate, we measured the slope of a linear fit to the normalized response rates for each of the temporally scaled repetitions of one input, for each neuron. For quasi-regular inputs, the average slope was 0.14 ± 0.06 (P > 0.1, mean ± SEM, n = 6 neurons), over a range of input rates varying from about one-third of each neuron’s unperturbed rate to three times the neuronal rate. For perfectly regular inputs, the slope was 0.11 ± 0.05 (P > 0.1, mean ± SEM, n = 6 neurons). Differences in slopes between responses to periodic and quasi-regular inputs were not statistically significant (P > 0.1, Student’s t-test). Responses to perfectly regular inputs were slightly faster overall than responses to quasi-regular inputs: the ratio of each cell’s overall mean rate in response to periodic stimuli to its mean rate for quasi-regular stimuli was 0.82 ± 0.04 (mean ± SEM, n = 6, P < 0.01, Student’s t-test).

Similar to the rate maintenance, despite the strong and varied inputs, cells maintained their individual C_V (Fig. 3B); while individual cells fired with different coefficients of variation, each cell tended to maintain its own C_V when driven with varying rates of input (cf. horizontal strata of various symbols in Fig. 3B). Input C_V had little overall effect on output C_V : responses to any given input (in Fig. 3B each input is represented by a different symbol) spanned almost the whole range of output C_V. Responses to perfectly regular inputs were slightly less variable than responses to quasi-regular inputs: the ratio of each cell’s C_V, in response to periodic stimuli, to its C_V for quasi-regular stimuli was 0.87 ± 0.02 (mean ± SEM, n = 6, P < 0.05, F distribution). Responses to periodic stimuli were, of course, still much more variable than the input itself.

As an additional means of testing the overall effects of our input on the output trains, we assessed the correspondence between input and output spike trains by the ISI-distance (Kreuz et al., 2007), which measures the ratio of input ICIs to output ISIs and is inverse to synchrony. Over the input speeds used, was minimal (synchrony was maximal) when input firing rates most closely matched the unperturbed firing rate of the neurons (Fig. 3C), in a form of spiking resonance. To measure the effectiveness of our stimuli in modulating spike times, we compared the value of to theoretical values computed for pairs of strictly periodic spike trains with a frequency mismatch (see Materials and methods). For very slow or fast inputs (normalized ICIs furthest from 1), the value of for our driven cells was smaller than 0.3, corresponding to more-similar trains. In comparison, mismatched periodic trains were less similar to each other and reached values of . We also noted that even for well-matched input rates (normalized ICI close to 1), reaches a minimum of ∼0.09 but does not reach zero, as pairs of periodic trains do; this indicates a residual effect of intrinsic variability in spike times despite the resonance to strong inputs. We also observed a resonance in reliability as a function of input C_V, though due to their construction and compression, our inputs did not sample the range of coefficients of variation sufficiently to quantify this effect.

Time-dependent effects

To investigate possible mechanisms for the maintenance of intrinsic firing rate, simultaneous with the resonance to input rate, we classified spikes according to their response signatures to the input (see Materials and methods) into categories of forced, perturbed and natural spikes. Spike classification used solely the response voltage traces to allow us to differentiate between spikes that were immediately forced by an input, spikes that followed an input with some latency, and naturally occurring spikes (Fig. 4A). To confirm the performance of our spike-classification algorithm, we then used the input to plot the latency of the classified spikes against the relative arrival time of inputs; indeed, spikes our algorithm classified as forced had uniformly small (< 0.1 periods) latencies (Fig. 4B), while the input-to-spike latencies of spikes we classified as perturbed were widely distributed in both their input-to-spike latency and their arrival time within the ISI.

Next we determined the effects of our input compressions on spike type by binning the classified output spikes in their respective categories, separated according to relative input rate (from fast to slow, relative to the unperturbed ISI of the neuron). From this separation, we saw one marked trend: the occurrence of natural spikes increased monotonically as input rate decreased (Fig. 5A). In this case, neurons spiked naturally in between input arrivals. Conversely, for fast inputs neurons spiked naturally much less often.

We also classified the inputs according to their success in eliciting an immediate spike into the categories of forcing, perturbing and neutral (see Materials and methods), and binned them similarly by relative input rate. Two effects from this separation emerged (Fig. 5B). We found the highest fraction of forcing inputs for the resonance case, where the rate of the input matched the rate of the unperturbed neurons; this is again largely intuitive. For the fastest inputs the fraction of neutral inputs is highest and, correspondingly, the percentage of inputs that successfully forced an immediate spike is decreased. This result indicates that neurons were overall less responsive to faster inputs, even while the neurons maintained their intrinsic rates of ∼5 Hz. For very slow inputs, the percentage of forcing inputs is enhanced by episodes of 1:2 synchrony.

One potential mechanism for rate maintenance despite strong input is via short-term memory, i.e. a dependence of any given ISI on the previous spike type (forced, perturbed or natural). In order to isolate the effect of single inputs from additional effects of subsequent inputs, we looked at the length of natural ISIs (which are undisturbed by input, and thus reflect intrinsic dynamics) following either a forced or a perturbed spike. Natural ISIs following a forced spike (e.g. Fig. 2B, second and second-to-last ISIs) were significantly longer (mean normalized ISI = 1.02, corresponding to an increase of ∼4 ms for a 200-ms theta cycle; P < 0.001) than following a perturbed spike (mean normalized ISI = 0.98), indicating a memory effect and a potential contributor to rate maintenance. Here we have corrected for the differences in arrival times of perturbed spikes, which arrive later after the previous input than forced spikes, as this could also affect the occurrence of natural spikes.

To visualize the temporal dependence of input influence during rate maintenance, we computed the probability of forcing inputs over input arrival times within the ISI (Fig. 6A; we will indicate this distribution in our model as the probability Q). As expected, inputs arriving very quickly after a spike are least likely to force another spike, and inputs arriving latest within the ISI are more likely to force immediate spikes, with a maximal probability at the natural ISI. In between these two extremes we observed a very pronounced peak in the probability of an input forcing a quick second spike, or doublet, centered at a normalized input arrival time of ∼0.07 of the natural ISI following a spike (this is 10 - 20 ms for a cell spiking at ∼5 Hz). Correspondingly, the probability of an input eliciting a perturbed spike was lower during this ‘doublet window’. Neutral inputs, which failed to elicit either a forced or perturbed spike, decreased in probability with elapsed time, except for a similar small dip corresponding to the doublet peak. We also looked at the latency distributions of perturbed spikes over different input arrival times [Fig. 6B; in our modeling, this will be the latency distribution R(Δ_p)]. The mean latency for perturbing inputs generally decreased with input arrival time, to a plateau latency of 0.3 (normalized); latency peaked during the doublet window.

Modeling

In order to test whether we could reproduce the neuron’s behavior using only these basic probabilities (cf. Fig. 6) as ingredients, we built a simple stochastic model that is schematically described in Fig. 7. We used two streams taken from our experiments: the input (ICIs), in exactly the same order as delivered during the experimental recordings (Fig. 8A); and neuronal ISIs, drawn randomly from the distribution P(t) of intervals recorded from unperturbed neurons (Fig. 8B). The model proceeded as follows:

We start with an output spike, at time t_out, and seek to determine the next output spike time t_out+1. Let t^e_out+1 denote the expected time of a possible next natural output spike, drawn from the distribution P(t). First we consider the next input, arriving at time t_in. If t_in > t^e_out+1, the input is too late; the next output spike, t_out+1 is taken as emitted at time t^e_out+1, and is considered of natural type. Alternatively, if t_in < t^e_out+1, an input arrives before a natural spike would have arrived; then the next output spike is taken as forced with probability Q(t_in − t_out) or as perturbed with probability 1 − Q(t_in − t_out) (Fig. 6A). In the former case, the forced spike t_out+1 is elicited at time t_in + Δ_f, where Δ_f is the measured latency of forced spikes (cf. Fig. 4B). In the latter case, the latency of a potential output spike Δ_p is chosen from the probability distribution R(Δ_p|t_in − t_out) (Fig. 6B, the latencies of perturbed spikes); the expected spike time t^e_out+1 = t_in + Δ_p is also compared with the time of the next input t_in+1, and forced or further delayed in a similar manner if t_in+1 arrives before the perturbed spike at t_in + Δ_p. Having thus determined the next spike time t_out+1, we then proceed from one output spike to the next until the experimental input stream has ended.

We compared the ISI distributions from the simulation and the experimental data. As shown in Fig. 8C, the distribution of simulated outputs reproduces the measured ISI distribution (black lines) reasonably well. Quantifying the overlap of the two distributions using Receiver-Operator-Characteristics (Hanley & McNeil, 1982) yields a value of 0.99, a highly significant similarity (P-value < 0.001). Our simulations reproduced the simultaneous rate maintenance (Fig. 8D) and the resonance effects of the experiments (Fig. 8E).

In the experimental data we distinguished three different input types (cf. 5, 6), while in the streaming model we can only distinguish a priori whether an input forces an immediate spike or not (Fig. 7, dashed green lines). The reason is causality: whether a not-forcing input subsequently becomes a neutral or a perturbing input depends on whether the next randomly drawn input appears before or after the next randomly selected output spike. However, for the simulation we can evaluate post hoc the relative fractions of perturbed and neutral spikes. In the simulation, the fraction of perturbing inputs was overestimated by only 12%, which, given the simplifying assumptions, serves as another indication that the model captures the essential features of the data very well.

In the temporal probabilities of input types (Fig. 6A) we noted a distinct peak in probability of forcing a spike near normalized t = 0.07 of the natural ISI. We hypothesized that this doublet peak would likely account for the slightly elevated probability of short ISIs in our simulations, and might account in part for the rate maintenance in output. To test this hypothesis, we reran our simulations after altering the probabilities of Fig. 6A: first, we artificially removed the doublet peak, by sending the probability of forcing a spike smoothly to zero for smaller elapsed arrival times (Fig. 6A, cyan line). As expected, this had the effect of diminishing the elevated probability of short ISIs in the output stream (Fig. 8C, cyan line). Second, we ran another simulation with temporally flattened response probabilities, where the response probabilities were set to their average over input arrival time (Fig. 6A, magenta). This model produced a larger number of short ISIs than found in the original data (Fig. 8C, magenta). Both altered models produced fewer long ISIs than were present in the original data. These are two simple examples of how such a model can be used to explore contributions of specific response attributes; many others are conceivable.

Removing the doublet peak also diminished or eliminated the rate maintenance and resonance effects. For both the experimental data and the output stream formed by simulating the experimental response properties, strong quasi-regular inputs varying in speed from one-third to three times the unperturbed neuronal rate had a slope of 0.14 (Fig. 8D). However, after smoothing away the doublet peak as above, the slope between input rate and output rate in our simulations was increased by 72%, to 0.24. In contrast, using flattened response probabilities resulted in a small decrease in slope, to 0.12. Moreover, the original streaming simulation matched the resonance feature of the experiments (Fig. 8E; the minimum in was 0.06 for the simulations), while both altered simulations failed to resonate to a single input speed. Thus, we conclude that the temporal response profiles of cortical cells (Fig. 6A) are finely tuned and shaped by the doublet window and by other contributing intrinsic effects (e.g. ionic and post-spike currents) to preserve the overall rate, while varying timings of individual spikes within a train in response to inputs.