A temperature rise reduces trial-to-trial variability of locust auditory neuron responses

Monika J. B. Eberhard 1,*,+, Jan-Hendrik Schleimer2,3,+, Susanne Schreiber2,3, Bernhard Ronacher 1,3

¹Humboldt-Universität zu Berlin, Department of Biology, Behavioural Physiology Group, Invalidenstr. 43, 10115 Berlin, Germany

²Humboldt-Universität zu Berlin, Department of Biology, Institute for Theoretical Biology, Philippstr. 13, 10115 Berlin, Germany

³Bernstein Center for Computational Neuroscience Berlin, Philippstr. 13, 10115 Berlin, Germany

⁺these authors contributed equally

Running head: Temperature effects on locust auditory neurons

*Corresponding author: M. J. B. Eberhard

E-mail address: <em>monika.eberhard@</em>uni-greifswald.de

Present address: General Zoology and Zoological Systematics, Zoological Institute and Museum, University of Greifswald, Anklamer Str. 20, 17489 Greifswald, Germany

Phone: +49 3834 86 4286

Fax: +49 3834 86 4089

Abstract

The neurophysiology of ectothermic animals, such as insects, is affected by environmental temperature, as their body temperature fluctuates with ambient conditions. Changes in temperature alter properties of neurons and, consequently, have an impact on the processing of information. Nevertheless, nervous system function is often maintained over a broad temperature range, exhibiting a surprising robustness to variations in temperature. A special problem arises for acoustically communicating insects, as in these animals mate recognition and mate localization typically rely on the decoding of fast amplitude modulations in calling and courtship songs. In the auditory periphery, however, temporal resolution is constrained by intrinsic neuronal noise. Such noise predominantly arises from the stochasticity of ion channel gating and potentially impairs the processing of sensory signals. Based on intracellular recordings of locust auditory neurons, we show that intrinsic neuronal variability on the level of spikes is reduced with increasing temperature. We use a detailed mathematical model including stochastic ion channel gating to shed light on the underlying biophysical mechanisms in auditory receptor neurons: due to a redistribution of channel-induced current noise towards higher frequencies and specifics of the temperature dependence of the membrane impedance, membrane-potential noise is indeed reduced at higher temperatures. This finding holds under generic conditions and physiologically plausible assumptions on the temperature dependence of the channels' kinetics and peak conductances. We demonstrate that the identified mechanism also can explain the experimentally observed reduction of spike timing variability at higher temperatures.

Keywords – temperature, grasshopper, auditory neuron, intrinsic variability

Introduction

Because physiological processes strongly depend on temperature all aspects of animal life are affected by it. Chemical reaction rates typically exhibit Q₁₀ values of 2.5-4 (Gewecke and Backhaus, 1995; Drosopoulos and Claridge, 2005), posing a challenge for ectothermic animals, such as insects, whose body temperatures are tightly coupled to the ambient temperature, and whose body functions usually have to be maintained over a broad temperature range of more than 20°C. For the nervous system of these animals, in particular, variations in temperature modulate fundamental properties of neurons resulting in changes of spike rates, conduction velocity, or transmitter release (Burrows, 1989; Montgomery and Macdonald, 1990; Janssen, 1992; Franz and Ronacher, 2002; Robertson and Money, 2012). As a consequence, the processing of sensory information as well as the coordination of movements should be affected. Nevertheless, several aspects of nervous system function of ectothermic animals have been found to show a relatively high level of robustness to temperature changes (eg. Roemschied et al. (2014); Rinberg et al. (2013); Tang et al. (2010); Caplan et al. (2014)) in spite of various temperature-induced modifications in their elements – features that we are currently only beginning to understand.

In this study, we investigate the effect of temperature on sensory processing in the auditory periphery of locusts with a focus on the variability of spiking responses. Earlier investigations on locust auditory receptors and interneurons revealed an improved temporal resolution at higher temperatures (Ronacher and Römer, 1985; Franz and Ronacher, 2002; Prinz and Ronacher, 2002). The neurophysiological data were complemented in behavioral tests: females of the grasshopper Chorthippus biguttulus are able to detect gaps as small as 1-2 ms in male songs at 35°C, whereas at 23°C larger gaps are necessary to allow detection (Helversen, 1972; Ronacher and Stumpner, 1988; Helversen and Helversen, 1997). A better temporal resolution might be due to faster deterministic dynamics at higher temperatures, but also a reduction in neuronal noise might account for these findings. However, it is currently not known how intrinsic neuronal noise is affected by temperature changes in these animals.

In the case of auditory receptor neurons, i.e. the cells in the first layer of the feed-forward network that constitutes the locust auditory periphery, the largest noise source is cell-intrinsic (as these neurons do not receive synaptic inputs from other cells). It is not obvious, whether intrinsic noise is reduced at higher temperatures. On the contrary, in view of larger peak conductances and shorter activation time constants of ion channels, one may even expect increased noise levels at higher temperatures. We hence intracellularly recorded the responses of identified neurons, exposing them to two different temperatures. Interestingly, trial-to-trial spike variability was consistently reduced at the higher temperature. To understand the mechanisms underlying this reduction of spike jitter at warmer temperatures, we followed a theoretical modeling approach: a detailed conductance-based model, including stochastic ion channels, was used to study the impact of temperature on spike rate and spike timing. Employing analytical techniques as well as model simulations we demonstrate that, indeed, voltage fluctuations at warmer temperatures are diminished in the vicinity of the firing threshold and, consequently, spike timing jitter is lowered. These findings hold under generic conditions and biophysically plausible assumptions on the temperature dependence of the channels' kinetics and peak conductances. The identified mechanism is likely to generalize beyond the locust receptor neurons.

Materials and Methods

Experimental animals and electrophysiology

Experiments were performed on adult male and female Locusta migratoria L., obtained from a commercial supplier and held at room temperature (22-25°C). Head, legs, wings, and gut were removed and the animals fixed with wax, ventral side down onto a Peltier element (3 x 1.5 cm), which was attached to a holder. The thorax was opened dorsally and the metathoracic ganglion subsequently stabilized on a NiCr platform. The whole thorax was filled with locust saline solution (Pearson and Robertson, 1981). Auditory receptors and interneurons were recorded intracellularly in the frontal auditory neuropil of the metathoracic ganglion using glass microelectrodes (borosilicate, GC100F-10; Harvard Apparatus, Edenbridge, UK) filled with a 3-5% solution of Lucifer yellow in 0.5 M LiCl. All electrophysiological experiments were conducted in a Faraday cage lined with foam prisms to minimize echoes. Neuronal responses were amplified (BRAMP-01; npi electronic GmbH, Tamm, Germany) and recorded by a data-acquisition board (BNC-2090A; National Instruments, Austin, TX, USA; sampling rate = 20 kHz).

The temperature of the preparation was controlled via the Peltier element connected to a 2V battery. A digital thermometer (GMH 3210, Greisinger electronic GmbH, Regenstauf, Germany) connected to a NiCr-Ni-thermoelement (GTF 300, Type K, Greisinger electronic GmbH, Regenstauf, Germany) was used to monitor and record temperature. The thermoelement was fixed between the Peltier element and the torso of the locust (underneath the ganglion), to prevent any disturbances of the neuronal recording. In most experiments, recordings were conducted first at 28-30°C, then the preparation was cooled down to 20-22°C and recordings were repeated for the same neuron at the low temperature. Temperature changes at the Peltier element were completed in approximately 1 min. When the low temperature of the Peltier element had been reached, another 2 min were waited before the second recording started. Tissue temperature at electrophysiological recording sites was derived from temperature measurements at the Peltier element and calibrated according to a calibration curve, which had been created previously by measuring temperatures with three thermoelements; one at the Peltier element, one directly at the inner side of the tympanum, and one at the ganglion (see Eberhard et al. (2014) and Roemschied et al. (2014)). While in the majority of the measurements the preparations were cooled down, in four preparations the temperature change was reversed, starting with a low temperature; the direction of temperature change had no influence on measured Q₁₀ values. After completion of the recordings Lucifer Yellow was injected into the recorded cell by applying hyperpolarizing current. The thoracic ganglia were then removed, fixed in 4% paraformaldehyde, dehydrated, and cleared in methyl salicylate. Stained cells were identified under a fluorescent microscope according to their characteristic morphology and physiology (Römer and Marquart, 1984; Stumpner et al., 1991). Altogether, we recorded from 13 receptor neurons, 25 local, and 24 ascending interneurons in 57 preparations (in 5 preparations, two neurons were recorded). Parts of the data on receptor neurons have already been used in a different context in previous studies (Eberhard et al., 2014; Roemschied et al., 2014) to compare the temperature dependence of neurons of all three processing stages, we chose to re-investigate these data. Among local interneuron types, two segmental, one bisegmental and two different T-neurons were recorded; among ascending neurons, 7 different neuron types were recorded (see Table 1).

Stimulation

To obtain spike rate-intensity-functions, we used acoustic broad band stimuli (100 ms duration, 1-40 kHz bandwidth) repeated 5 times each at 8 intensities, rising from 32 to 88 dB SPL. Acoustic stimuli were stored digitally and delivered by a custom made program (LabView 7 Express, National Instruments, Austin, TX, USA). After a 100 kHz D/A conversion (BNC-2090A; National Instruments, Austin, TX, USA), the stimulus was routed through a computer-controlled attenuator (ATN-01M; npi electronic GmbH, Tamm, Germany) and an audio amplifier (Pioneer stereo amplifier A-207R, Pioneer Electronics Inc., USA). Acoustic stimuli were unilaterally broadcast by speakers (D2905/970000; Scan-Speak, Videbæk, Denmark) located at ± 90° relative to the longitudinal axis of the preparation, at a distance of 30 cm from the animal. To control for directional sensitivity of a recorded neuron, the sound stimulation was first played from the left and then the right side or vice versa. Sound intensity was calibrated with a half inch microphone (type 4133; Brüel & Kjær, Nærum, Denmark) and a measuring amplifier (type 2209; Brüel & Kjær, Nærum, Denmark), positioned at the location of the preparation.

To test for changes in intrinsic variability for more complex stimuli, an additional set of stimuli was used in 15 recordings (see Fig. 5, Table 1,2): two model songs containing rectangular syllable envelopes, corresponding to mean syllable-pause lengths measured in Chorthippus biguttulus male songs at 30°C and 20°C (Block1 and Block2), part of a natural song recorded from a Ch. biguttulus male singing at 30°C (Origsong1), and the same natural song, expanded 1.7 times to correspond to a natural song at 20°C according to Helversen (1972) (Origsong2). All stimulus envelopes were filled with a broadband noise of 1-40 kHz bandwidth. Song stimuli were presented 8 times each at 70 dB SPL, usually from the side (left or right) which showed a more sensitive reaction during presentation of the 100 ms stimuli. Note that locusts do not use calling songs for mate attraction, nevertheless, physiology and morphology of the auditory peripheral neurons are highly evolutionary conserved and homologous for L. migratoria and Ch. biguttulus (Neuhofer et al., 2008).

Data analysis

Spike times were extracted from the digitized recordings by applying a voltage threshold. Mean spike rates were calculated to obtain spike rate-intensity-functions per neuron, stimulation side, and temperature. From these curves, temperature coefficients (Q₁₀ values) of the firing rate were determined: Rate changes with a 10° temperature shift can be expressed by the temperature coefficient Q₁₀:

Q10=(XY)10(Tx−Ty)

where X is the rate at higher temperature (Tx) and Y is the rate at the lower temperature (Ty). A mean Q₁₀ value was subsequently calculated for each neuron using only values at intensities eliciting spike rates above a threshold set at 50% of the maximum spike rate at the high temperature (see Fig. 1A). In addition, Q₁₀s for first spike latencies, as well as action potential duration (width) and amplitude (height) were calculated for each recorded neuron. To determine Q₁₀s for action potential width and height, spontaneous action potentials recorded during trials (before the start of a stimulus) were detected, superimposed and the mean action potential shape was calculated. From this, height and width at half of the action potential amplitude were measured for each neuron and temperature (see Fig. 2D). Significance of differences between high and low temperature for the various characteristics measured was estimated using Wilcoxon matched pairs signed-rank tests, significance of differences in Q₁₀ values of the three processing stages was calculated using Kruskal-Wallis tests, and post-hoc pairwise comparisons were performed using Wilcoxon rank-sum-tests, Bonferroni corrected.

To estimate intrinsic variability (trial-to-trial variability) of spike responses for each neuron and temperature, the pairwise metric distance between spike trains of the five repetitions per stimulus intensity (or between the eight repetitions of the model songs) was calculated according to Rossum (2001). This metric yields an intuitive measure for the dissimilarity of spike trains, with large distance values indicating a high dissimilarity, that is, a large trial-to-trial variability. The van Rossum metric allows one to vary the temporal resolution of the comparison (via a resolution parameter τ). Equipped with very large τ values (> 200 ms), the metric largely ignores differences in the timing of spikes, and spike train dissimilarity depends only on spike count differences. With τ values of a few milliseconds, differences between the spike trains in both spike count and spike timing contribute to the dissimilarity. Previous investigations showed that the timing of spikes plays an important role for the encoding of acoustic signals at the level of thoracic neurons (Stumpner et al., 1991; Franz and Ronacher, 2002). Therefore we used a temporal resolution of τ = 5 ms, which has been found to sufficiently encompass the coding properties of auditory neurons at different processing stages (Machens et al., 2003; Wohlgemuth and Ronacher, 2007; Neuhofer et al., 2008). In addition, we also performed an analysis with τ varying between 2 and 1024 ms.

The van Rossum distance also depends on the number of spikes (Neuhofer et al. 2011). In order to compare distance values between hot and cold recordings, which may differ in spike rates, we standardized the distance values by the square root of the number of spikes elicited during the stimulus, to get a comparable average distance (see appendix 1). Subsequently, one mean distance value per neuron was calculated (mean distance over all intensities where a spike response was elicited, and a mean distance per neuron for all model songs together, respectively). For the model songs, the first 150 ms of each recording were omitted, to analyze the responses in the adapted state only.

All analyses were done using Matlab (R2012a, The MathWorks); graphs were edited in CorelDraw (X6, Corel Corporation).

Simulation

The modeling approach was focused on the first processing stage, the receptor neurons. A model capturing mechanotransduction as well as spike generation was implemented. It had previously been fitted to the locust mechano-receptors (Fisch et al., 2012) and replicates experimentally measured inter-spike interval distributions and serial correlations. For the transduction step, sound pressure waves evoke tympanal vibrations that in turn open and close mechano-sensitive ion channels (appendix 2, Eq. (7)). The parameters of the tympanal oscillator are based on experiments using laser vibrometry (Schiolten et al., 1981). Biophysical details of the model are described in appendix 2 (see Fig. 6A, for an equivalent circuit). All model parameters were taken from Fisch et al. (2012) and can be found in Table XX. Only the transduction parameters xbase, α and ks were adapted to fit our measured rate intensity curves. The temperature coefficients were chosen from realistic ranges (Hille, 1992): Q₁₀ values of the peak conductances ranged in the interval [1,2], while the Q₁₀ values of the ion channel kinetics are between [2.5,4]. The exact values were chosen such that in that the temperature effect on the firing rate was comparable to that of the measured responses (see Fig. 6B).

The model was simulated at 22°C and 32°C. For the stimuli, s(t)=20×10Idb(t)/20ξs(t), the broad band carrier that is typical for Grasshopper songs was described by a Gaussian white noise process ξs(t). Two classes of stimuli were applied: (i) A white noise carrier with constant amplitude Idb(t)=const. given in dB SPL, to determine the rate-intensity curves (the result can be seen in Fig. 6B). (ii) The same time-dependent amplitude modulation, Idb(t), as in the natural songs used in experiments (Fig. 6C). Stochastic ion channels were approximated by a diffusion equation instead of simulating the channel’s Markov models (Linaro et al. 2011). More details can be found in appendix 2. The simulated spike trains were analysed in the same way as the experimentally recorded ones. All simulations were performed in the brian2 library (Stimberg et al., 2014). The temperature compensated response in the firing rate was achieved by selecting the Q₁₀ values with a genetic algorithm (GA) from their given ranges. The GA objective was to minimized the mean squared-difference between hot and cold response, but was only allowed to choose realistic Q₁₀ values from their respective intervals. For this the deap toolbox (Fortin et al., 2012), written in python, was used.

Analysis

To support the simulations and deepen the insight into the temperature effects on neuronal noise, formulas for the statistics of current and voltage fluctuations are provided in the following. Their detailed derivations can be found in in appendix 3. In particular, the dependence on the temperature susceptibilities, Q₁₀ values, is highlighted in the analysis. Peak conductances, ion channel kinetics and reversal potentials depend on temperature, T in Kelvin, as follows:

gk(T)=gk(Tbase)Q10(gk)(T−Tbase)/10,τ̃ki(T)=τ̃ki(Tbase)Q10(τ̃i)−(T−Tbase)/10,Ek(T)=Ek(Tbase)TTbase.

Here, the index k stands for elements of the set, k∈K, of all channel types. In our model these were K={Na,K,M,R,L}, see Fig. 6A. The time constants τ̃ki, together with the steady-state activation curves, which are not temperature dependent, describe the gating kinetics of channel k. Depending on the complexity of the underlying Markov model, there may be several time constants per channel involved.

To obtain a concise description of the total membrane current fluctuations in the receptor neuron model, it is approximated by a single coloured, Gaussian noise process ηˇ, with correlation function cov(Δ)=σˇ2(v,T)e−∥Δ∥/τˇ(v,T). For a clamped voltage v and a fixed temperature T, the total noise power (i.e., the integral over the noise spectrum) is given by (cf. Appendix 3)

σˇ2(v,T)=∑k∈K∑i=1Mk−1σki2(v)gk2(T)(Ek(T)−v)2.

The number of states in the channel's Markov model is denoted by Mk. The noise variance σki, associated with each channel state is defined in Linaro et al. (2011) and depends only on the steady-state activation curves, and hence not on temperature. With this approximation the temperature influences the total noise power only through the temperature susceptibilities of the peak conductances, gk(T), and the reversal potentials, Ek(T). The effect on the reversal potentials is not substantial (~3%). The Q10(gk) values, are similar to that of aquatic diffusion, and hence small compared to those of the reaction rates. In total, σˇ has a weak temperature dependence. In contrast, the time constant of the equivalent noise process ηˇ, is given by

τˇ(v,T)=σˇ2(v,T)∑k∈K∑i=1Mk−1gk2(T)(Ek(T)−v)2σki2(v)/τki(v,T).

Only in some case, like a simple two states channel or the linear chain in the model's K+ channel, τki∝τ̃ki, otherwise the expressions for the τki's may involve several of the original τ̃ki's (cf. Linaro et al. (2011), Tab. 2). In general, the temperature scaling of τˇ involves all Q₁₀ values, including the strong temperature dependencies of the reaction kinetics. With this, the spectrum of the total noise current at a clamped command voltage v is then approximately a Lorentzian

PI(f)=τˇ(v,T)σˇ2(v,T)1+(fτˇ(v,T))2.

The quantitative link between spike jitter and subthreshold membrane voltage fluctuations is nontrivial (Alijani and Richardson, 2011), yet voltage noise is a better predictor than unfiltered current fluctuations. Therefore, the current spectrum in Eq. (4) should be filtered by the membrane impedance to obtain the voltage fluctuations

σV2=∫dfZ(f)PI(f).

The impedance for our model can be approximated by (cf. appendix 3 for a justification)

Z(f)≈1(1c∂If∂V)2+f2.

Note that the steady state value of ∂If/∂V, again, only depends on the Q10 parameters of the peak conductances, gc(T), and the reversal potentials, Ec(T). Consequently, the impedance is affected by temperature to a lesser degree, as can be inspected in Fig. 7B.

Results

The first three processing stages of the auditory pathway form a hierarchically organized feed-forward network (in the metathoracic ganglion) and comprise receptor neurons, thoracic local interneurons, and ascending interneurons, respectively (Stumpner et al., 1991; Stumpner and Ronacher, 1991; Vogel and Ronacher, 2007; Clemens et al., 2012). First, we characterized effects of temperature on response characteristics of neurons across all three layers. To this end, each individual cell was recorded at two different temperatures (at approximately 20°C and 30°C).

Temperature effects on basic parameters of neuronal responses

In all 62 recorded neurons spike rate increased with higher temperature (Fig. 1, Table 1). The general shape of the spike-rate-intensity functions did not change with temperature (i.e. saturating or unimodal curve, see Fig. 1), nor was the basic spiking pattern of cells (phasic versus tonic response) affected. On average, the temperature dependence of the spike rate was smallest in receptor neurons (mean Q₁₀ of 1.38 ± 0.19, median: 1.33). For local and ascending interneurons the temperature effect was more pronounced (Fig. 1D; mean: 2.45 ± 1.48 (median: 1.88), and 1.96 ± 0.91 (median: 1.68), respectively), see also Table 1.

Temperature changes affected action-potential shape and first-spike latencies (Fig. 2). At all three processing stages, neurons exhibited a significant decrease in action potential width (Fig. 2E,H) and latency with increased temperature (Fig. 2F,I), whereas spike amplitudes showed no consistent changes (Fig. 2D,G). Q₁₀ values for action potential height and width did not significantly differ between neurons at the three processing stages (Kruskal-Wallis test, AP height: χ²= 2.33, P = 0.31, AP width: χ²= 1.31, P = 0.52).

Temperature effects on intrinsic neuronal variability

Timing of spikes is thought to contribute to the neuronal representation of fast amplitude modulations in grasshopper acoustic communication signals [Machens et al. (2001);Wohlgemuth et al. (2010)). Trial-to-trial variability of spiking responses can hence impair the processing of these vital signals.

We therefore quantified the trial-to-trial variability of responses using the spike train metric introduced by Rossum (2001). The metric can be applied with different values of the time constant τ (which is a parameter to the metric), setting the timescale of spike train comparison (small τ emphasizing spike timing on short timescales, large τ shifting emphasis rather to spike count than spike timing). Across cells from all processing stages of the peripheral network, spike train distances were significantly larger at the lower temperature, provided we used a high temporal resolution (τ = 5 ms; Fig. 3, Wilcoxon matched pairs signed rank tests). In other words, for both the short 100 ms acoustic stimuli (Fig. 3) and the longer model songs (Fig. 5A-C) spike timing variability was decreased at the warmer temperature.

For receptor neurons, in particular, this relation was observed across all values of τ used in the analysis (Fig. 4A). Hence, both the intrinsic variability of spike count as well as of spike timing was decreased at warmer temperatures. For low values of τ, local and ascending interneurons exhibited the same trend, i.e. the spike timing variability decreased at higher temperature (Fig. 4B,C). However, with larger τ values (above 32 ms) differences in variability between the cold and warm temperatures disappeared, indicating that at these processing stages spike count variability was less affected by a temperature change than spike timing variability.

These results are further supported by the data obtained with long song models (Fig. 5). With a focus on spike timing, using a τ = 5 ms, most neurons exhibited a larger intrinsic variability at the lower temperature (Fig. 5A,C). Similar to the short acoustic stimuli, for small values of τ spike timing variability was on average larger at the low temperature whereas this effect disappeared or even reversed for larger τ values (Fig. 5D-E).

Modeling

To identify the mechanisms underlying the observed reduction in intrinsic variability at elevated temperatures, we turned to computational modeling. Temperature is known to increase the peak conductances of ion channels. One may hence surmise a rise in conductance noise and, consequently, in current noise. Yet, two additional factors come into play: also transition rates between channel states are expedited, and the translation of current changes into voltage is governed by the membrane impedance, whose temperature dependence is likely to further modify the temperature effect on voltage noise. To dissect the impact of temperature on noise fluctuations at the different levels of current noise, voltage noise, and spike timing jitter, we analyzed a previously published quantitative model of the receptor dynamics (Fig. 6A).

First, a temperature dependence was introduced to this model assuming Q₁₀ values for peak conductances and reaction rates of the individual ion channels, see Eq. (1) (and Table 3). Q₁₀ parameters were chosen from realistic parameter ranges to obtain rate-intensity curves with a temperature dependence comparable to that observed in the experimental data (Fig. 6B). For details on the chosen parameter set see appendix 2. Next, model responses to natural song stimuli were obtained from simulations at two different temperatures (Fig. 6C). The analysis of spike variability in the model yielded the same relation as in the experimental recordings: spike train variability was reduced at the warmer temperature (compare Fig. 6D and Fig. 4A).

An advantage of this approach is that the model allows for an explicit dissection of the underlying biophysical mechanisms, which we discuss in the following. For moderate firing rates, spike jitter depends on voltage fluctuations at threshold (Alijani and Richardson, 2011). The reduction in spike jitter should, therefore, be accompanied by smaller voltage fluctuations at the warmer temperature. In order to understand how the current noise produced by the multitude of stochastic ion channels affects voltage fluctuations, the model's noise current spectrum and membrane impedance were obtained both through simulation and analytical techniques (see Eq. (4) and appendix 3). While an increase in temperature entailed only moderate changes to the variance of the total membrane noise current, it redistributed the noise power in the current spectrum to higher frequencies (Fig. 7A). However, the membrane impedance, which “translates” current noise to voltage noise, was much less affected by temperature and exhibited low-pass filter characteristics with virtually identical cut-off frequencies at both temperatures (Fig. 7B). Combining these two facts yielded the explanation for the reduction of neuronal noise at warmer temperatures: The additional power at high frequencies of the current noise was not translated to voltage fluctuations, because the impedance values at these high frequencies were low (at both temperatures). In contrast, the reduction in low-frequency power of the current noise also resulted in a lower contribution of these frequencies to voltage noise at the warm temperature so that the overall voltage noise was reduced. Mathematically, the different behaviour of total noise power and impedance cutoff in contrast to noise current cutoff can be understood from the formulas summarised in materials and methods; for a derivation see appendix 3. From these expressions it is clear that only the noise cutoff frequency, 1/τˇ(v,T), is influenced by the strong temperature susceptibilities of the reaction kinetics. In contrast, the total noise strength, i.e., the integral of the power spectrum, and the membrane impedance depend on the Q₁₀ values of the peak conductances only (appendix 3, Eq. (2) and (3)). The stronger temperature dependence of the cutoff frequency is a direct consequence of the fact that kinetic Q₁₀ values are usually larger than the Q₁₀ values of peak conductances. While the former Q₁₀ values are in the range of 2.5-4 (Hille, 1992), the latter lie close to the Q₁₀ values of aquatic diffusion coefficients, i.e., << 2.

Discussion

The most remarkable finding of this study is a decrease of the overall trial-to-trial variability of auditory neuron responses at higher temperatures (Figs. 3 and 5). Several other response characteristics important for neuronal signaling were influenced by temperature in a similar way as reported for other auditory neurons; for example action potential width and spike latencies decreased with rising temperature (Figs. 1 and 2, compare with Abrams and Pearson (1982), Coro et al. (1994), Fonseca and Correia (2007), Korsunovskaya and Zhantiev (2007)). In view of the strong temperature influence on ion channels kinetics (Q₁₀ ~ 2.5-4, Hille (1992)), the observed decrease in spike jitter at first glance seemed counterintuitive.

Mathematical modeling explains the effect of temperature on noise

We searched for a mechanistic explanation of the observed variability reduction, by studying voltage fluctuations near threshold in a model introduced by Fisch et al. (2012). We demonstrated that increasing temperature redistributes current noise power to higher frequencies, which then are filtered out by the membrane impedance and hence contribute little to voltage variance (Fig. 7). Together this explains the voltage noise reduction at warmer temperatures. The change in the spectrum of the total noise current is also in accordance with the formulas given for single channel noise spectra (DeFelice, 1981; O’Donnell and Rossum, 2014). Our analytical treatment revealed under which conditions of the ion channels' Q₁₀ parameters the reduction in voltage noise found in the simulations will take place. The total current noise power as well as the impedance cutoff are affected by the typically small Q₁₀ values of peak conductances, while the cutoff frequency of the noise spectrum depends on the much larger Q₁₀ values of opening and closing reaction rates (Hille, 1992), and consequently show a stronger increase with temperature. Based on numerical simulations, voltage fluctuations near threshold have previously been described in conductance-based models, where a qualitatively similar trends for temperature change was observed (Steinmetz et al., 2000).

In addition, the model was stimulated with the naturalistic songs used in the measurements. The experimentally observed reduction in trial-to-trial variability was reproduced by the model. We hence established that the model not only complies with the experimental spike statistics in a constant stimulus paradigm at a single temperature [Fisch et al. (2012)), but also agrees with data for naturalistic time-dependent song stimuli at different temperatures. Our analytic formulae can be applied to models containing arbitrary ion-channel combinations, in order to analyze their impact on the noise statistics. In future work, it will enable us to detect compensatory mechanisms between the large amount of existing ion channels with their different time scales and temperature susceptibilities. While it has been demonstrated experimentally that ion channels, like Kv1 potassium channels, have an impact on cortical spike variability (Higgs and Spain, 2011), our quantitative understanding of how the potpourri of channels present in nerve membranes jointly impact the voltage fluctuations is still incomplete. The mathematical ansatz in appendix 3 can support a more rigorous analysis in this direction.

Relevance of temperature and noise

The body temperature of grasshoppers is tightly coupled to the ambient temperature, and their body functions have to be maintained over a broad temperature range of more than 20°C. In particular, the neuronal processing of calling and courtship songs is crucial for the recognition and attraction of sexual partners and is strongly influenced by temperature (Helversen, 1972, 1979; Bauer and Helversen, 1987; Ronacher and Stange, 2013). The songs of many grasshopper species comprise fast amplitude modulations which constitute important signals for species recognition and sexual selection (Elsner, 1975; Helversen, 1979; Ronacher and Römer, 1985; Ronacher and Stumpner, 1988; Kriegbaum, 1989). Hence, elements of the auditory pathway should attain a high temporal resolution to allow for a robust evaluation of courtship signals – and spike train variability is detrimental for this task (Neuhofer et al., 2010; Ronacher, 2014).

Grasshopper auditory receptor neurons encode the fine structure of amplitude modulations of sound stimuli, in their instantaneous firing rate (Machens et al., 2001; Wohlgemuth et al., 2010). The lower intrinsic variability at warmer temperatures hence could allow for a better resolution of fine temporal details. This view is supported by studies using a modulation transfer function paradigm that demonstrated an increased temporal resolution of auditory receptor neurons at higher temperature (Ronacher and Römer, 1985; Franz and Ronacher, 2002; Prinz and Ronacher, 2002). Indeed, grasshoppers tested in a behavioural gap detection paradigm were able to detect gaps as small as 2 ms at 30°C, whereas at 22°C the minimal detected gap width was approximately 4 ms (Ronacher and Stumpner, 1988). Similar improvements of temporal resolution with increasing temperature have been found in the fly’s visual system (Warzecha et al., 1999; Tatler et al., 2000).

In summary, and elevated temperature leads to a reduction of spike timing variability in the grasshopper auditory periphery, which can be explained by a net decrease in the impact of channel noise on membrane voltage. The mechanisms, characterized in the mathematical model, are likely to generalize and to apply to neurons beyond the specific system at hand. This study shows that in order to decipher the effect of temperature on neuronal computation and to understand principles that enable a robustness to temperature changes (Robertson and Money, 2012; Rinberg et al., 2013; Caplan et al., 2014; Roemschied et al., 2014), not only deterministic, but also stochastic mechanisms need to be taken into account. Further studies must reveal how the interplay of several different temperature mediated effects leads to the robust encoding of auditory signals, which is reflected in the behavior of grasshoppers that respond to acoustic signals within a large temperature range.

APPENDIX

Appendix 1: Normalisation of the van Rossum metric

The original van Rossum metric (Rossum, 2001) is defined by d̃2=∫([y1(t)−y2(t)]*h(t))2dt, in which two Dirac-spike trains are denoted as y1,2(t)=∑i=1Nsp(1,2)δ(t−tk(1,2)), with spikes occurring at times tk(1) and tk(2). The convolution operator is denoted by * and h(t) is either an exponential kernel, h(t)=H(t)2/τe−t/τ (Rossum, 2001), or the synaptic α-function (Machens et al., 2003). H denotes Heaviside's step function: H(t)=1 if t>0 and H(t)=0 if t<0.

In this article, a normalized version of the van Rossum metric is used. The square of the original expression is divided by the spike count, Nsp, to obtain

d2=2d̃2Nsp(1)+Nsp(2)=2Nsp(1)+Nsp(2)∫([y1(t)−y2(t)]*h(t))2dt.

The motivation for the normalization is to obtain a measure by which cold and hot spike train variabilities can be put into perspective, even if the spike counts are different. Fig. 8 shows that the unnormalised van Rossum distance d̃ between two Poisson processes, indeed, scales with the square root of the mean spike count. The figure shows an example with τ=2, but note that the graph is independent of the time scale τ. The same scaling is found for two inhomogeneous Poisson processes with the same instantaneous rates and, generically, for all Poisson processes in the limit of small τ (Tomas and Sousa, 2008).

Appendix 2: Auditory receptor model

The current balance equation for the voltage, v(t), of a single compartment model of the auditory receptors in Locusts, proposed in (Fisch et al., 2012), reads

cv̇+INa+IK+IL+IM+IR=0

It comprises the ionic currents INa, IK and IL, responsible for the spike generation, as well as an M-type adaptation current, IM, and the receptors' transduction current, IR. The membrane capacitance c is 1μF/cm2. All parameter values can be found in Table 4.

The tympanal deflections, x(t), induced by an external sound pressure wave, s(t)=20×10Idb(t)/20ξs(t), are described as a damped stochastic oscillation

ẍ+(2/τd)ẋ+ωty2x=αs(t)+σtyξx(t).

The input s(t) is imbibed with a gain factor α, that was determined from stroboscopic measurements (Breckow and Sippel, 1985; Fisch et al., 2012). The thermal noise of the tympanum has variance σty2=(4αkBT)/(τdB), where B is the vibrating area of the tympanum and kB is Bolzmann's constant. The eigenfrequency and the damping time constant of the tympanum, ωty and τd, respectively, have previously been determined by laser vibrometry (Schiolten et al., 1981; Eberhard et al., 2014).

The tympanal vibrations are transduced into a receptor current IR=GR(t)(v−ER) by an unidentified receptor, possibly of the TRP family as in Drosophila (Zhang et al., 2013). The receptor channel is assumed to have two states, follows first order kinetics, and has an activation function described by (Howard and Hudspeth, 1988; Hudspeth et al., 2000)

z∞(x)=11+exp(−kskBT(x−xbase))+11+exp(kskBT(x+xbase)).

The spring constant ks, the half-maximum of the mechano-transducer xbase, and the tympanal gain α, were adapted to fit the recorded rate-intensity curves, cf. Fig. 6B. All other channels follow typical Markov state models from the literature (Fisch et al., 2012).

In total, there are three different noise sources in the equations above. First of all, thermal noise from the tympanum. Secondly, the input which consists of a modulated white noise carrier. And thirdly, noise from stochastic ion channel gating, discussed next.

Stochastic ion channels

Instead of simulating the master equations for the stochastic ion channels, one (Linaro et al., 2011) of several (Fox and Lu, 1994; Goldwyn et al., 2011; Orio and Soudry, 2012) diffusion approximations available in the literature is used. This casts the problem into a stochastic differential equation (SDE), which facilitates the following mathematical analysis.

As an example, take the receptor channel from above. It is modeled as a two-state (open-closed) Markov model (Fisch et al., 2012). The gating variable GR that approximate the two-state Markov dynamics, can be split into a deterministic z and a stochastic ηz part, GR=g‾R(z+ηz). According to Linaro et al. (2011) the governing equations read

τz(T)ż=(z∞(x)−z),τz(T)η̇z=−ηz+2τzσzξz(t).

In the two state case, the noise variance given by Fox and Lu (1994) is valid. It reads σz2=τz(z∞(x)(1−z)+(1−z∞(x))z), which in the steady state becomes

σz2(x)=NR−1z∞(x)(1−z∞(x)).

For fixed tympanal position x, Eq. (8) is an Ornstein-Uhlenbeck (OU) process. The temperature dependence of τz(T) follows Eq. (1).

Ion channels with n>2-dimensional state space require as many OU processes to represent their correlation structure, as there are reactions. The voltage dependent activation curves and time constants of these OU processes are obtained from an eigendecomposition of of the infinitesimal transition matrices of the Markov models (Tuckwell, 1989). For the Na⁺ and K⁺ channels used here, they are given in (Linaro et al., 2011). The M-type adaptation conductance is modeled as a two-state channel like the receptor.

Appendix 3: Model analysis

The complete rate-intensity curve of a nonlinear and stochastic membrane is not analytically tractable. The calculation of the statistical properties of the firing rate can be divided into three mathematical problems: (i) well below threshold, the firing rate is given as an escape process, with Poisson statistics; (ii) well above threshold the system exhibits a stochastic limit cycle, with inverse Gaussian statistics; (iii), an intermediate regime where both descriptions break. The intermediate regime is important for amplitude coding, as it spans the largest dynamic range. The spike fluctuations in that regime are related to voltage noise (O’Donnell and Rossum, 2014). Hence, the following calculation focuses on deriving the voltage noise close to threshold under varying temperature.

Simplification of the transduction

To start with, the transduction dynamics are simplified. This step reduces the simulation time, because the integration of the tympanal oscillations of the full system would render the SDE system stiff. Furthermore, the transduction channel can now be treated as just one more channel that contributes to the total voltage noise.

Note from the model parameters in Table 4 that the difference in time scale between the eigen-frequency of the tympanum and the time constant of the receptor channel, τz>2π/ωty, prevents a locking of spikes to the tympanal oscillations, in accordance with experimental observations (Sippel and Breckow, 1983). Moreover, this difference in time scale allows for an adiabatic elimination (Titulaer, 1980) of the fast variables x and ẋ, leaving us with analytic expressions for mean and variance of the receptor gating variable, z, as a function of the input intensity Idb. This procedure formalises the intuition that, because the receptor kinetics are slower than the tympanal dynamics, they are only affected by the average statistics of the deflections. Thus, it suffices to take the equilibrium distribution, p∞(x;Idb), of the tympanum's vibrations to average over Eq. (8). Since Eq. (7) is a noisy harmonic oscillator, the marginal equilibrium distribution of the tympanal excursion, x, is given by

p∞(x;Idb)=(2πσx)−1exp(−x22σx2).

The dependence on the input Idb is mediated by the variance

σx2=((α⋅20×10Idb/20)2+σty2)τd/4ωty2,

i.e., a louder sound produces a larger deviation in the tympanal deflections. The theoretical distributions and the histograms obtained from numerical simulations at two input intensities are illustrated in Fig 8A.

Applying the adiabatic elimination procedure (Titulaer, 1980) to the drift and diffusion parts of Eq. (8) requires calculation of the averaged steady-state activation curve

z̃∞(Idb)=∫−∞∞dxp∞(x;Idb)z∞(x)≈1−tanh(ksxbase/(2kBT))(1+(ksσxkBTsech(ksxbase2kBT))2)12.

To obtain the analytical approximations above, the activation function z∞ is approximated by a Gaussian, so that the integrals can be evaluated using Laplace's method (MacKay, 2003).

In addition, the averaged diffusion coefficient is needed. So, mutatis mutandis, z∞(1−z∞) can be approximated by one minus a Gaussian, which yields

σ̃z2(Idb)=∫−∞∞dxp∞(x;Idb)σz2(x)≈2eβxbase22(1−βσx2)(14−1(1+e2ksxbase/kBT)2)NR1−βσx2,

where

β=−ks2(5+40e2ksxbase/kBT+18e4ksxbase/kBT+e8ksxbase/kBT)2(3+4e2ksxbase/kBT+e4ksxbase/kBT)2kB2T2.

The reduced gating SDE for the two state receptor channel then has the following structure

τz(T)ż=z∞(Idb)−z+2τzσz(Idb)ξt.

In this approximation, for any fixed mean input Idb and temperature T, the transduction gating variable will be Gaussian. This effectively eliminates the state space dimensions corresponding to the noisy tympanal oscillations. The accuracy of this approximation can be inspected in Fig. 9B and C.

Noise current spectrum

The currents in the balance equation for the voltage dynamics can be separated into deterministic, Id(v), and stochastic, If(v), parts

Cv̇+Id(v)+If(v)=0.

The drift contains the deterministic model components. Near the fixed point it reads

Id(v)=g‾Na(T)m∞3(v)h∞(v)(v−ENa(T))+g‾K(T)n∞4(v)(v−EK(T))+g‾L(T)(v−EL(T))+g‾M(T)w(v−EM(T))+g‾R(T)z̃∞(Idb)(v−ER(T)),

while the stochastic part is

If(v)=g‾Na(T)(v−ENa(T))∑k=1MNa−1ηk+g‾K(T)(v−EK(T))∑j=1MK−1ηj +g‾M(T)(v−EM(T))ηw+g‾R(T)(v−ER(T))ηz.

Here, Mc is the number of states of the c∈{Na,K,M,R}-channel. The noise processes are exponentially correlated OU processes

τk(v,T)η̇k=−ηk+2τk(v,T)σk(v)ξk.

The analytic expressions for the voltage dependent variances σk(v) and time constants, τk(v,T) are determined from the eigenvectors and eigenvalues of the infinitesimal transition matrix of the Markov chains respectively and can be found in (Tuckwell, 1989; Linaro et al., 2011). The expression for the variance of the receptor channel was derived in Eq. (11).

As a next step, the fluctuating part is approximated by a single OU process, fitted to reproduce the correlation time of the superposition of OU processes in If. The auto-correlation of If under voltage clamp and in steady state is

⟨If(t)If(t+Δ)⟩=g‾Na2(v−ENa)2∑k=1MNa−1σk2(v)e−∥Δ∥/τk(v) +g‾K2(v−EK)2∑j=1MK−1σj(v)2e−∥Δ∥/τj(v)+g‾M2(v−EK)2σM(v)2e−∥Δ∥/τM(v)+g‾R2(v−EK)2σ̃R2(Idb)e−∥Δ∥/τz.

This is approximated by a single OU process, ηˇ with correlation σˇ2(v,T)e−∥Δ∥/τˇ(v,T), by Taylor expanding both the single OU and the sum of OUs. Based on this expression both σˇ and τˇ can be identifed (see Eqs. (2) and (3)).

Hence, the power spectrum of the total noise current is the Fourier transform of the correlation function at a clamped command voltage v is then approximately a Lorentzian Pi(f)=τˇ(v)σˇ2(v)1+(fτˇ(v))2.

Importantly, the total variance of the membrane current, σˇ2, is only affected by the Q10 values of the peak conductances, which are relatively small (<2.0). On the other hand, the cutoff frequency, τˇ−1, of the total membrane noise current depends on the Q10 parameters of reaction kinetics, which are comparatively large (>2.0). This leads to the redistribution of noise power to higher frequencies shown in Fig. 7A.

Membrane impedance

The subthreshold impedance filter can be derived by linearisation around the resting potential. The poles and zeros of the impedance filter are then related to the eigenvalues of system's Jacobian matrix. In the class of a type I neuron this simplifies and the impedance is a lowpass filter approximately given by

Z(f)≈1((1c∂If∂V)2+f2).

Note that the steady state value of ∂If/∂V only depends on the Q10 parameters of the peak conductances. Consequently the impedance it affected by temperature to a lesser degree as can be inspected in Fig. 7B.

Together with the conclusion of the previous section this means: If all Q10(gc) are well below 2 and Q10(τi) sufficiently large, voltage noise fluctuations will be reduced with increasing temperature (cf.: Fig. 7C).

Acknowledgements

We thank J. Clemens for providing important input and help concerning Matlab programming, and F. Roemschied for providing a tool to calculate mean action potential shape.

Grants

This work was funded by the German Research Council (DFG: SFB 618, grant to BR and SS, RO547/12-1) and the Federal Ministry of Education and Research, Germany (01GQ1001A, 01GQ0901, 01GQ0972, grants to BR and SS).

Disclosure

No conflicts of interest, financial or otherwise, are declared by the authors.

Figure legends

Fig. 1. Examples of spike rate-intensity-functions obtained from locust auditory neurons at two different temperatures. A: local interneuron SN1, B: ascending interneuron AN1, C: ascending interneuron AN3 (for spike rate-intensity-functions of receptor neurons see Eberhard et al. 2014; Roemschied et al. 2014). D: Boxplots of Q₁₀ values of spike rate of all recorded cells showing means (circles), medians (thick lines), 25/75 percentiles (boxes), 1.5 Inter-Quartile-Range (whiskers) and outliers (+); horizontal bars show pairwise comparisons (Wilcoxon signed-rank-tests, Bonferroni corrected). *** – P < 0.001, NS – not significant.

Fig. 2. Effect of temperature on characteristics of auditory neurons. A-C show voltage traces of a receptor (A), a local (B), and an ascending interneuron (C) at high and low temperatures in response to a 100 ms broadband noise stimulus (56 dB SPL). D-F: Boxplots of Q₁₀ values for action potential height (D), width (E), and first spike latency (F). G-I: Pairwise comparisons of action potential height (G), width (H) and latency time (I) at the two temperatures (hot & cold). P values shown here were calculated using a Wilcoxon matched pairs signed rank test for each group. NS – not significant.

Fig. 3. A-C: Spike-raster-plots of example neurons recorded at high and low temperatures using short broad-band noise stimuli at different intensities (indicated at the sides of the raster-plots, dB SPL). Five repetitions per stimulus intensity are shown for a receptor neuron (A), a local interneuron BSN1 (B) and an ascending interneuron AN3 (C). D: Variability values (metric distance in arbitrary units) calculated using the metric according to van Rossum, with τ = 5 ms. E-F: Pairwise comparisons of distance values at the hot vs. cold temperature for receptors (E), local (F) and ascending (G) interneurons (IN).

Fig. 4. A: Variability values at cold temperature subtracted from variability values at hot temperature for receptors, local and ascending interneurons for different values of the metric's temporal resolution parameter τ. At large τ values only spike count differences contribute to the metric value, while for small τ also differences in spike timing become relevant. Grey boxplots show a significant difference from zero (i.e. a significant change of intrinsic variability with temperature), while white boxes are not significantly different from zero (i.e. no change of variability with temperature).

Fig. 5. Variability values resulting from applying the van Rossum-metric to spike trains of neurons recorded during stimulation with four model songs. A: Variability values at hot temperatures minus values at cold temperatures for τ = 5 ms. Grey diamonds mark the mean variability calculated over all four model songs, grey circles indicate mean variability differences obtained with the short 100ms stimuli. B: Example spike-raster-plots of a receptor neuron and ascending interneuron AN3, recorded at high and low temperatures; shown are eight repetitions of the same stimulus (Origsong1) . C: pairwise comparisons of mean variability values for τ = 5 ms. D,E: Mean change (variability hot – variability cold) of intrinsic variability with temperature, using different values of τ. For each neuron, an average variability value for all four model songs was calculated.

Fig. 6. A: Equivalent circuit of the auditory receptor model. Tympanal vibrations are transduced by receptor channels (ER). Other channels included in the model are spike generating Na-, and K channels, leak channels (L) as well as adaptation channels (M). B: Rate-intensity curves of the model at 22°C and 32°C. Also included in B is a plot of spike with at two temperatures as derived from the model. C: Spike response for natural, time-dependent stimuli at 22°C (cold) and 32°C (hot). D: Average difference in spike train-metric (hot minus cold) for different values of τ. In all cases the distance value decreased at the high temperature, indicating both a decrease in spike-time jitter and spike count variability. Model and simulation parameters can be found in appendix 2.

Fig. 7. A: Power spectral density of the total current noise at 22°C (light grey, dashed line) and 32°C (dark grey, solid line) in voltage clamp condition (command voltage is -67mV). All stochastic ion-channels, including the transduction contribute to the fluctuations, but to different degree. The weighted sum in Eq. (2) and (3) describes this behavior. Dots show estimated spectra, while lines are the theoretical prediction from Eq. (4) in the Appendix. The main effect of the temperature increase is a shift of noise power to the high frequency range. Vertical line denotes the cutoff frequency of the impedance filter. B: Temperature dependence of the membrane impedance filter in the subthreshold dynamical regime. Simulations (dots) are compared with the approximative, analytical formula (lines, Eq. (6)). C: Membrane voltage fluctuations near the resting state are reduced in the model neuron with higher temperature. The observed temperature dependence is a combination of the redistribution in noise power towards higher frequencies in (A) and the relative temperature invariance of the cut-off frequency of the membrane impedance in (B). Parameters as in Fig. 9.

Fig. 8. Unnormalized spike train metric, d∼, for two independent Poisson processes with increasing number of spikes. Solid line shows the theoretical square-root scaling of the spike metric and dots are simulated Poisson processes. Nsp – spike count.

Fig. 9. Comparison between theory and simulation for the receptor model with transduction. A: Probability distribution of tympanal deflections (as indicated in the schematics below) in response to white noise stimuli at two different sound intensities (Idb = 40 dB SPL and Idb = 50 dB SPL). Solid lines represent the Gaussian distribution from Eq. (9). B: Probability distributions of the conductance fluctuations of the receptor channels (depicted in the schematics - R) induced by the two inputs. Solid lines represent theoretical predictions based on the quasi-static averaging, Eqs. (10) and (11). C: Comparison of the mean and variance of the receptor gate z as a function of the sound input. Black dots depict the mean open probability and errorbars denote the standard deviation. The dark grey line shows the theoretical curve from Eq. (10) and the grey shaded area the standard deviation, see Eq. (11). D: The intensity and fluctuations of the transduction current are translated into a spike frequency by voltage dependent ion channels (depicted in the schematics below), resulting in a rate-intensity curve. The two output frequencies for Idb = 40, 50 dB are marked by vertical lines. T=22°C, all other parameters as given in the Tables 3 and 4.

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