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Investing integrator frequency response curves

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The peaks seen at high firing rates coincide at a time of approximately —3 ms from the second spike. The firing rate of a Purkinje cell changes depending on modulation of its inputs [31] — [35]. For example, during locomotion in cats the firing frequencies of Purkinje cells can increase from an average of about 40 Hz [34] to more than Hz [35].

We first recorded at the spontaneous firing frequency, and if the spontaneous frequency was low, we next increased the firing rate by injecting a positive constant current. Alternatively a negative constant current was injected if the spontaneous frequency was high. The PRCs for both fast and slow states were calculated Fig. This change in the PRC was reversible, as when the neuron was allowed to relax back to its intrinsic firing rate 40 Hz the PRC returned to a square shape Fig.

When the neuron was then allowed to fire at its intrinsic firing rate 84 Hz the sharp peak in the PRC reappeared Fig. Therefore, the switch in Purkinje cell dynamics reflected in the switch of the PRC can also occur in the same cell. B Injection of depolarizing current into the cell in A increases its firing rate Hz and the PRC switches to being phase-dependent. C When the depolarizing current is removed, the cell relaxes back to its intrinsic spontaneous firing rate 40 Hz and its PRC switches back to being phase-independent.

E When hyperpolarizing current is injected into the cell to decrease its firing rate 26 Hz , the PRC switches to being phase-independent. F Removal of the hyperpolarizing current returns the cell to its intrinsic spontaneous firing rate 84 Hz and its PRC switches back to being phase-dependent. We have shown that the traditional method for calculating PRCs results in a bias, particularly in neurons exhibiting high ISI variability. We developed a corrected method for calculating PRCs which removes most of this bias.

Our method can be directly applied to noisy experimental data. We used this corrected approach to measure for the first time the PRCs of Purkinje cells at various firing rates. This suggests that Purkinje cells can behave as perfect integrators at low firing rates, which has important consequences for our view of the integrative properties of these neurons. We have determined Purkinje cell PRCs by injecting brief current pulses and measuring the phase change in the subsequent neuronal spiking.

Since at the typical spontaneous firing rates of Purkinje cells these phase changes were small compared to the spike jitter during spontaneous spiking [1] , many trials were required. This revealed a general bias of the traditional method at late phases of the PRC in the presence of noise Fig. We characterized the effect in a model with and without noise, and showed that the bias is related to inhomogeneous phase histograms caused by interspike interval jitter Fig.

To correct for this, we developed a new method, which recovers periodicity in the spike jitter due to noise Fig. We showed that this method homogenizes the phase sampling in the experimental data and removes most of the bias observed in the PRCs calculated using the traditional method Fig.

Our corrected approach can be directly applied to existing experimental data in order to measure PRCs under low signal-to-noise conditions. It should be applicable to a wide range of cell types, as neuronal noise and the resulting ISI variability are not restricted to Purkinje cells [36]. The use of indirect methods to obtain PRCs, for example from the spike triggered average [23] or the poststimulus time histogram PSTH [24] are possible alternatives to the traditional method.

Here we have applied a correction to the traditional method, which resulted in reliable PRC measurements in Purkinje cells. Further alternative methods for calculating PRCs exist. For example, dynamic clamp was previously used to study hippocampal spike-timing-dependent plasticity in relation to PRCs [37]. In this special case, underlying subthreshold oscillations provide phase locking. Such a method is only applicable if phase information is accessible to the experimenter, independent of spiking.

PRCs can also be calculated using Bayesian statistics [25] , or by injecting trains of rectangular current pulses [38] and noisy inputs [11]. These methods result in periodic PRCs, but only because periodicity is imposed as part of the optimization fitting techniques employed.

In conclusion, our method can be applied to noisy experimental data to calculate PRCs while avoiding possible bias or overfitting problems present in some of the currently available methods. A wide, comparative study will be required in the future to find out which methods for calculating the PRC yield the best results under different conditions. Purkinje cells fire spontaneously and modulate their firing in response to synaptic input.

For example, the rate of Purkinje cell firing can exhibit a consistent temporal relationship with wrist movement [31] or be monotonically related to eye velocity during smooth-pursuit eye movements [40]. How is the integration of single inputs affected by the firing rate of the Purkinje cell?

We have addressed this question by measuring the PRC at different firing rates. Using our new approach, we determine experimentally the PRCs of cerebellar Purkinje cells and show that their shape changes significantly depending on the firing rate compare Fig. To the best of our knowledge, this is the first study to report a phase-independent PRC in a mammalian neuron. It was previously reported in a spike-frequency adaptation model of cortical neurons that an increase in firing frequency causes a shift of the PRC peak from rightward skew to the centre with a decrease in amplitude [24] , implying that the integrative properties of this model neuron change depending on the firing rate.

Specifically, it was suggested that the model cell acts like a coincidence detector at low firing rates and more of an integrator at higher firing rates [24]. Purkinje cells appear to show the opposite behaviour, acting as perfect integrators at low firing rates. The shape of the PRC is thought to be linked to the type of excitability of the neuron.

Neurons with type I excitability, whose f-I curves are continuous, are thought to display purely positive PRCs while neurons with type II excitability, characterized by a discontinuity in the f-I curve at the onset of firing, exhibit biphasic PRCs [11] , [13] , [14]. While biphasic PRCs intuitively result in resonator behavior, neurons with purely positive PRCs act as integrators of synaptic input [11] , [13] , [14] , [43].

Although Purkinje cells exhibit type II excitability [2] , [44] , [45] , their PRCs are positive at all firing rates, implying that they are integrators rather than resonators. These findings suggest that the type of excitability of a neuron is not strictly correlated with the PRC shape. Similarly, Tateno and Robinson [15] showed that low-threshold spiking, fast spiking and non-pyramidal regular spiking interneurons can exhibit both purely positive and biphasic PRCs which do not always strictly correspond to the type of excitability of the neuron.

The shape of the PRC has functional implications for the integration of synaptic inputs. At high firing rates, Purkinje cells are most sensitive to inputs during the last 3 ms of their firing cycle Fig. It has been shown theoretically that oscillators which are described by type I PRCs and are coupled by excitatory synapses tend not to synchronize [16]. However, the opposite is true for inhibitory coupling between oscillators [16] , [46] , such as coupled Purkinje cells. Indeed, theoretical and experimental evidence indicates that Purkinje cells tend to synchronize via inhibitory inputs [4] , [6] , [7].

As the firing rate of Purkinje cells decreases, and the levels of synaptic and intrinsic conductances and currents are modified, the PRC switches from monophasic to phase-independent Fig. The phase-independent PRCs at low firing rates suggest that Purkinje cells integrate their synaptic inputs independently of their timing within the interspike interval Fig.

Our results therefore support the idea that at low firing rates, Purkinje cells cannot read out the timing of their inputs, which would exclude the use of a temporal code. Instead, in this regime they are well suited for rate coding. What are the biophysical mechanisms responsible for the switch in PRC behaviour at different firing rates? To generate an entirely flat PRC would require a neuron to effectively completely compensate for its leak conductance.

This is illustrated by the example of the PRC of a simple leaky integrate-and-fire neuron in which the leak conductance was eliminated Fig. S1B and C. However, this absence of leak is unlikely to occur in real Purkinje cells, and the biophysical implementation remains unknown.

PRCs qualitatively similar to those observed in our experiments at high firing rates can be generated by the Purkinje cell model of Khaliq and colleagues [26] Fig. However, when the firing rate is lowered in the model, no qualitative switch in the shape of the PRC can be observed. However, none of these models fully capture the experimentally determined switch in Purkinje cells, perhaps reflecting the fact that both of these models represent dissociated Purkinje cells.

Thus, our experimental results could aid the refinement of existing models in order to capture the full dynamic behaviour of Purkinje cells. In conclusion, our experimental findings indicate that Purkinje cells display different dynamic behavior depending on their firing rate. At high firing rates these neurons act as coincidence detectors of synaptic inputs, with maximal sensitivity at the late phases of the interspike interval.

In contrast, at low firing rates Purkinje cells are not suited for precise coincidence detection, but instead appear to perfectly integrate their inputs independently of their position within the interspike interval. Thus, at high firing rates Purkinje cells can transmit information via a temporal code whereas at low firing rates they are well-suited for rate coding. Thick-walled, filamented, borosilicate glass electrodes Harvard Apparatus Ltd. Purkinje cell somatic whole-cell patch-clamp recordings were obtained using an internal solution containing the following in mM : methanesulfonic acid, 10 HEPES, 7 KCl, 0.

All recordings were performed at Series resistance and pipette capacitance were carefully monitored and compensated throughout the experiment. To determine how spike timing during spontaneous firing is shifted by a brief perturbation, we injected rectangular current pulses of 0.

A control PRC cPRC was calculated using the unperturbed part of the voltage traces and assuming a current pulse injection 0 pA amplitude after 50 ms 25 ms of spontaneous firing in subsequent trials of ms ms for a slowly rapidly firing cell.

The dynamics of a neuronal oscillator can be reduced to a single variable: the phase. Depending on the phase of the stimulus, a change in phase, , of subsequent spiking will occur. Traditional method: A brief current pulse is injected at a random time. The spikes before and after it are identified.

When the unperturbed is defined as the mean ISI , a point on the PRC plot becomes: 1 where denotes the ISI which contains the brief current pulse and is the PRC point calculated in reference to the spike just prior to the stimulus. The resulting curve is a plot of against.

The curve is positive negative when the injected current advances delays the next spike. A moving average was calculated with a Gaussian kernel over the raw data. Corrected method: A major problem with the traditional method is the loss of periodicity of the sampling reference Fig.

In order to restore periodicity, points unaffected by the stimulation pulse can be added to the ensemble of PRC points, which allows the spiking jitter to average out properly. These points can be obtained from the same data by adding PRC values when the preceding ISI is taken into account: 2 When preceding and subsequent ISIs are taken into account as in: 3 and 4 periodicity in the spiking jitter is restored, phases are sampled homogeneously and the cPRC becomes flat.

In the resulting plot, the phase interval ranges from to and the PRC component affecting directly the interval corresponds to all points in the phase interval [0,1], termed PRC 1. Peak-to-baseline ratio: In order to distinguish the phase-independent PRCs from the phase-dependent ones, PRCs were classified according to the peak- to-baseline ratio. The peak-to-baseline ratio is then defined as:.

The noise injection resulted in a coefficient of variation of ISIs of 0. Current pulses of 0. Data shown is taken from more than trials. Additional neuron models were used in the supplementary parts of the manuscript. For Fig. S1 , the Morris-Lecar model was directly implemented using parameters from [28]. The parameters for the leaky integrate-and-fire model were: a membrane time constant of , a reset potential of , a threshold potential of , a membrane resistance of , and a steady driving current of to result in 50 Hz firing and was simulated at time steps of.

For the non-leaky integrate-and-fire model the time constant was set to infinity and , otherwise the same parameters were used. An alternative model for Purkinje cell firing was used for Fig. In this model, current pulses of 0. Simulation results were analysed in the same way as the experimental data. Validation of the corrected method. A Comparison between the traditional method red line and the corrected method black line to obtain PRCs and their numerical green line and analytical blue dashed line no-noise pendants using the example of the Morris-Lecar model for which the analytical PRC can be calculated by the adjoint method.

Curves have been rescaled to their maxima to aid comparison. In all cases the corrected method performs better than the traditional method. The use of the corrected method is particularly important in B which corresponds best to the case observed in the experimental data from Purkinje cells at low firing rates. Purkinje cells therefore act as perfect non-leaky integrators at low frequencies compare Fig. This suggests that Purkinje cells act like leaky integrators at high frequencies compare Fig.

Comparison of the corrected and traditional methods for obtaining PRCs. The population averages obtained with the traditional method thick green lines are qualitatively different from the population averages obtained using the corrected method thick black and red lines. The bias is such that the conclusions obtained in this study would not have been possible without developing the new method. PRCs in different model neurons.

A PRCs obtained with the model of Khaliq et al. However, the PRCs at low firing rates left are still not flat. Performed the experiments: EP. Analyzed the data: EP HC. Abstract Cerebellar Purkinje cells display complex intrinsic dynamics. Author Summary By observing how brief current pulses injected at different times between spikes change the phase of spiking of a neuron and thus obtaining the so-called phase response curve , it should be possible to predict a full spike train in response to more complex stimulation patterns.

Introduction Cerebellar Purkinje cells exhibit a wide range of dynamical phenomena. Results A bias in the traditional method for calculating PRCs Somatic whole-cell patch-clamp recordings were made in current-clamp mode from spontaneously firing Purkinje cells in mouse cerebellar slices.

Download: PPT. Figure 1. Purkinje cell PRCs determined using the traditional method. Figure 2. Interspike interval variability causes a bias in the traditional method to calculate PRCs. Improving the traditional method to obtain PRCs in the presence of noise Our new method to correct for the bias in the traditional PRC and obtain a homogeneous phase histogram is illustrated in Fig.

Figure 4. Validation of the corrected method for obtaining PRCs. A frequency-dependent switch in Purkinje cell dynamics Spontaneous firing frequencies of Purkinje cells range from 10— Hz both in vitro [1] , [2] , [4] , [5] and in vivo [3] , [30]. Figure 5. Two types of PRCs depending on the Purkinje cell firing rate. Figure 6. A frequency-dependent switch in Purkinje cell dynamics. Figure 7.

The switch in PRC shape can occur within the same cell. Discussion We have shown that the traditional method for calculating PRCs results in a bias, particularly in neurons exhibiting high ISI variability. A new approach for determining PRCs We have determined Purkinje cell PRCs by injecting brief current pulses and measuring the phase change in the subsequent neuronal spiking.

Purkinje cell dynamics depend on firing rate Purkinje cells fire spontaneously and modulate their firing in response to synaptic input. Functional implications The shape of the PRC is thought to be linked to the type of excitability of the neuron.

Materials and Methods Ethics statement All procedures were approved by the U. Home Office. Supporting Information. Figure S1. Figure S2. Figure S3. References 1. Neuron — View Article Google Scholar 2.

J Physiol — View Article Google Scholar 3. Nat Neurosci 8: — View Article Google Scholar 4. Nat Neurosci — View Article Google Scholar 5. J Neurosci — View Article Google Scholar 6. View Article Google Scholar 7. View Article Google Scholar 8. Reyes AD, Fetz EE Two modes of interspike interval shortenings by brief transient depolarizations in cat neocortical neurons. J Neurophysiol — View Article Google Scholar 9. Winfree AT Phase control of neural pacemakers. Science — View Article Google Scholar Guevara MR, Glass L, Shrier A Phase locking, period-doubling bifurcations, and irregular dynamics in periodically stimulated cardiac cells.

Izhikevich EM Dynamical systems in neuroscience: the geometry of excitability and bursting. Oxford: MIT press. Scholarpedia 1 12 : Izhikevich EM Neural excitability, spiking and bursting. Int J Bifurcation Chaos — Neurocomputing — Tateno T, Robinson HPC Phase resetting curves and oscillatory stability in interneuron of rat somatosensory cortex. Biophys J — You are in a professional report. Information for entry level users report for a newbie The automatic selection of the optimal AMP models.

Information for all Versus of characteristics Compare the sensitivity and impedance. Quick jump to the schedule in this report:. Quick jump to the schedules in this pro report: ZS10 Pro Test report - general data. About this report This is a headphone report for professionals. You can see a simple report for "dummies" with explanations of the main characteristics for KZ ZS10 Pro.

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This chapter examines the four basic negative feedback connections that might be used with an operational amplifier, and details the action of. Your integrator has two 15M resistors and an 8pF capacitor. have actually calculated both values from the desired frequency response. › operational-amplifier-as-differentiator.