Fig 1.
Circuit involved in the production and modulation of complex spikes.
(A) Simplified scheme of the inputs to the inferior olive (IO). Sensory input reaches the IO directly from the brainstem and spinal cord. In our study, we used facial whisker input that is relayed via the sensory trigeminal nuclei (TN). This input is considered the “sensory input” in our modeling studies. The IO also receives continuous inputs from other brain regions, which we modeled as the “contextual input”. The contextual input consists of excitatory input from the cerebral cortex (e.g., the motor (M1) and somatosensory cortex (S1)), either directly or relayed via the nuclei of the meso-diencephalic junction (MDJ), as well as of inhibitory input from the cerebellar nuclei (CN). The output of the IO is directed via its climbing fibers to the Purkinje cells (PCs) in the cerebellar cortex and via its climbing fiber collaterals to the CN. Sensory input also affects the contextual input indirectly, via the strong pathway from the TN via the thalamus (TH) to the cerebral cortex. (B) Representative trace of Purkinje cell activity showing simple spikes (as downward deflections) occurring at a high frequency and occasionally complex spikes (CS; marked with a blue dot). A part of the trace is enlarged in (C). All recordings were made in awake mice.
Fig 2.
Oscillatory dynamics in complex spike firing in vivo.
(A) Representative inter-complex spike-interval (ICSI) histogram of spontaneous firing of a representative Purkinje cell, together with the convolved probability density function (in blue). The shades of the bins represent the normalized oscillation strength. Note that the peak at 0 ms is removed to improve visibility. (B) Heat map showing the normalized oscillation strengths of 52 Purkinje cells, which were ordered by the time to the first side peak. The arrow in B indicates the cell illustrated in A. (C) The peak times of the ICSI distributions during spontaneous firing, as observed in B, against the Z-scores of these peaks. A Z-score <3 was taken as sign for the absence of rhythmicity (light blue fillings). The pie diagram shows the fraction of Purkinje cells with rhythmic complex spike firing (Z-score > 3; dark blue fillings). (D)-(F) The corresponding plots for the 46 (out of the 52) cells that showed a significant complex spike response following whisker air puff stimulation. Complex spike rhythmicity is displayed using peri-stimulus time histograms (PSTHs) aligned on the peak of the first response and ordered based on the latency to the second peak.
Fig 3.
Similarity between spontaneous and sensory-induced rhythmicity.
For each of the 30 oscillating Purkinje cells that were recorded in both the presence and absence of sensory stimulation, we compared the timing of the maximum of the first side peak in the inter-complex spike interval (ICSI) histogram in the absence of sensory stimulation (x-axis) with the timing of the difference between the maxima of the first and second peak in the peri-stimulus time histogram (PSTH; y-axis) following sensory stimulation. Color and size of data points indicate amplitude (in Z-score) of the ICSI peak and of the interval between first and second response peaks of the PSTH, respectively. Grey circles indicate the data-points falling in the low Z-score (<3) group during spontaneous firing. Note that colored circles are preferably located around the identity line.
Fig 4.
Stability of rhythmic complex spike activity in a Purkinje cell of an awake mouse.
(A) Rhythmic complex spike (CS) responses to whisker pad air puff stimulation in a representative Purkinje cell recorded in crus 1. Representative trial from an extracellular recording with a stimulation frequency of 0.5 Hz. In this and subsequent panels, CSs are marked with symbols representing their order of occurrence following the stimulus within each trial. (B) Rhythmic behavior of CSs becomes apparent in a raster plot. Note that the rhythmicity is relatively stable over the 1,000 trials. (C) Same as in B but with trials sorted by the latency to first CS. This reveals that the occurrence of CSs is largely organized in temporal windows of opportunity. Some CSs appear in later response peaks without prior firing in earlier windows of opportunity. As the CS response may appear in the early and/or in the late window, this is suggestive of a readout from an underlying oscillatory process. This phase dependence would not appear if it would be solely due to a reset followed by transient oscillation evoked exclusively in the spiking cell. If CS firing during the later peaks was predicated on an earlier phase-aligned CS, then later CS responses would have no reason to align with the second or the third peaks of the PSTH (in D and E). (D) PSTH of CS firing. The bin-width is 10 ms and the blue line shows the convolved histogram with a 5 ms kernel. (E) The same PSTH as in D, but now the probabilities of the first, second and third CS after the stimulus shown separately, highlighting the occurrence of windows of opportunity for stimulus triggered CSs.
Fig 5.
A tissue-scale network model of the inferior olive.
(A) To study the relative impact of the different components of the inferior olive we employed a biologically realistic network consisting of 200 model neurons embedded in a 3D grid of 10 x 10 x 2. The model neurons received input from two sources: a phasic input dubbed "sensory input" and a continuous fluctuating current, emulating a "contextual input". The “sensory input” reaches a subset of neurons synchronously and is modeled as an activation of glutamate receptor channels with a peak amplitude of 5 pA/cm2. The “sensory input” is complemented with a continuous input that is a mixture of excitatory and inhibitory synapses representing signals from different sources (see Fig 1A). The “contextual input” is modeled as an Ornstein-Uhlenbeck noise process delivered to all neurons in the model network, with a 20 ms decay, representing temporally correlated input. The model neurons are interconnected by gap junctions. (B) The model neurons display subthreshold membrane oscillations (STOs), represented by the lower circle, and spiking, represented by the upper circle. The colors indicate which current dominates which part of the activity pattern. An Ih current (green), a hyperpolarization-activated cation current, underlies a rise in membrane potential; the IT current (red), mediated by low threshold T-type Ca2+ channels (Cav3.1), further drives up the membrane potential and from here either the subthreshold oscillation continues (lower circle) or a spike is generated (upper circle), mediated through INa and IK currents (yellow and blue, respectively); both after a spike or a subthreshold peak, influx of Ca2+ through the P/Q-type type Ca2+ channels (Cav2.1) leads to a hyperpolarization due to Ca2+-dependent K+ channels (blue); this hyperpolarization in turn will activate the Ih current resuming the oscillation cycle. A resulting trace of the STO punctuated with one spike is diagrammatically displayed on the right. For a full description of currents, consult methods. (C) 3D connectivity scheme of the 200 neurons in the network model. Links indicate gap junctional interconnections; the color of each neuron in the model represents the number of neurons to which that cell is connected. The thick green lines indicate first order neighbors of the center cell. (D-E) Membrane potential of cells in the absence of input, for networks with and without gap junctional coupling. (F-G) Cell activity sorted by amplitude, indicating that coupling the neurons of the network brings non-oscillating cells into the synchronous oscillation. (H-I) Distribution of STO frequency and amplitude for coupled and uncoupled model networks.
Fig 6.
Synaptic input strongly affects the oscillatory behavior in a network model of the inferior olive.
(A) A phasic and synchronous input mimicking sensory input (see Fig 5A) was delivered to a subset of cells (indicated in red) of the model network. (B) Membrane potentials of all 200 cells in the network model in the absence of contextual input. Red trace (top) displays membrane potential average of cells receiving sensory input (see masks in A). Below are the membrane potentials of the model cells as a heat map. Each row exhibits the membrane potential of one model cell with the small circles indicating spikes. In the absence of contextual input, model neurons displayed long lived, synchronous and regular transient subthreshold oscillations. Note that the fact that the two stimuli fall in similar phases is coincidental. (C) Patterns of spiking responses changed dramatically in the presence of contextual input. Whereas in the absence of contextual input model neurons showed synchronous subthreshold oscillations and were silent except upon receiving sensory input, in the presence of contextual input the subthreshold oscillations are irregular and not strictly synchronous, while spikes occur throughout the simulation (~1 Hz average network firing). Note that despite the largely uncorrelated (90%) contextual input, model cells did not fire homogenously as in B, though loose clusters of synchronously firing cells did emerge. (D) Rhythmic responses to stimulation are reproduced by the inferior olivary network model. A representative cell of the model network with responses to sensory input reproduces rhythmic features of the PSTH and an auto-correlogram as found in vivo (see Fig 4). Raster plot of olivary spiking triggered by 182 stimuli, ordered by latency to the first spike following the stimulus. As seen in the in vivo data, the first spike after the stimulus can fall in one of multiple windows of opportunity. For this simulation, we included the contextual input. (E) Peri-stimulus time histogram showing the response peaks upon sensory input. (F) Raster plot like in C, but with overlaying membrane potentials. For clarity only the first 50 stimuli are displayed. Spike symbols follow conventions as in D. Trials with low latency were preceded by a dip in the preceding membrane potential, a phenomenon that can also be observed in D and H. (G) Average membrane potential of this neuron aligned by spikes that were either due to sensory input (87 spikes; red) or produced spontaneously (648 spikes; blue). The light shaded backgrounds represent the 10% and 90% percentile ranges of membrane potential. Dips in membrane potentials preceding the spikes reveal that for both groups a prior inhibition increased the probability of spiking. Importantly, except for a refractory period, we did not observe a clear oscillation pattern following either spike type, due to variability imposed by the contextual input. (H) shows the difference in spike triggered membrane potential averages for stimulus triggered and "spontaneous" spikes. (I) Auto-correlograms comparing rhythmicity of spikes produced during stimulation with spikes due to background (spontaneous) activity. Periodic stimuli delivered to the network reduced rhythmic responses at the peak. The extreme short-latency responses (in the center of the auto-correlogram) are due to the spikelets detected in the model traces that can also be seen in D.
Fig 7.
Synaptic input determines the phase-dependent gating properties of the inferior olive network model.
(A) Phase-dependent spiking was prominent in our network model of the inferior olive in the absence of contextual input. We repeatedly stimulated the same 54 model cells with synchronous sensory input (cf. Fig 5A). The other 146 neurons of the network model were present, but not directly stimulated. Phase-dependent gating was studied using tandem stimulation with varying inter-stimulus intervals. The membrane potential of each of these 54 model neurons is plotted in A as a waterfall plot and in B as a heat map. The simulations are grouped by the interval between the two stimuli, starting with a single stimulation (red), and continuing with two stimuli with small to large intervals. In 13 equal steps two whole cycles of the subthreshold oscillation were covered. (C) Average and inter-quartile range (shaded area) of membrane potentials of neurons in the network that were only perturbed once at 0 ms (red) or twice (other colors). The phase of the subthreshold oscillations of the “synchronized” cells in the network was used to compute the data displayed in D and E. Note that the average membrane potential at the perturbation at 0 ms only went up to -15 mV and not up to the average spike peak of approximately 10 mV, because in the population only a fraction of neurons spiked. (D) Spike probability plot as measured for the second perturbation at the different phases of the membrane oscillation. Spike probability was computed as the number of neurons firing divided over the total number of neurons getting input. The first perturbation given at the peak of the oscillation is demarked as the start of the oscillation cycle. Perturbations at both a half and one-and-a half cycle (π and 3π, respectively) did not trigger spikes, whereas perturbations at either one or two full oscillation cycles (2π and 4π, respectively) did (see also A and B). The repetition of the ‘sinusoidal’ probability curve shows that gating is not (only) due to a refractory period, but follows the hyperpolarized phase of the oscillation (not shown). (E) Phase response curve showing that perturbations early in the oscillation phase did not have a large impact on the timing of the spike, but halfway the oscillation the perturbations advanced the ongoing phase considerably, with impact declining linearly to the end of the full cycle. (F) Gallop stimuli provide indirect evidence for an underlying oscillatory process. This idealized diagram illustrates the impact of a resetting stimulus on the future spiking probabilities. Here we assume that resetting spikes occur only during the rising phases of the STO (red) and not during the falling phases (blue). One can query for the impact on spiking probability for any given initial phase at any time after the first stimulus. Different intervals lead to biased response ratios. (G) In order to test the impact of the presumed phase of the inferior olivary neurons, we applied a “gallop” stimulation pattern, alternating short (250 ms) and long (400 ms) intervals (top row). “Sensory input” (vertical red shaded bars) were delivered and spikes were counted in response windows (RW) 0–180 ms post-stimulus. Response probabilities were compared between response windows (RW) in short (empty bars) and long intervals (filled bars). Many cells in the model prefer either the short or the long window (for spiking cells, see left lower panel). However, in the presence of contextual noise, this preference largely washes off, seen as approximately symmetric responses in each of the intervals. Asterisks indicate that a tiny fraction of cells still displayed significant preference for particular intervals, which could be attributed to the low number of spikes fired by these cells.
Fig 8.
Gap junctions reinforce synchronous behavior after sensory stimulation.
Impact of periodic stimulation on network phase in the presence of contextual stimuli for networks with (left column) and without gap junctions (right column). Brown bars indicate stimulation time. (A) Stimulus triggered membrane potential for all cells in the network (n = 100 stimuli). Stimulation causes a peak in the membrane potential of the directly stimulated cells (red line; cf. mask in Fig 5A). The effect of the current spreading through gap junctions is visible in the cells immediately neighboring the stimulated cells (green line), whereas it is absent in the network model without gap junctions. (B) Increase of synchrony in the network due to a single stimulus. Synchrony is displayed as phase concentration over time (the Kuramoto parameter) for cells under the stimulation mask (colored) and other cells (gray). Stimulation generated a similar synchrony peak in both networks, but the transient synchrony after the peak is considerably higher and longer lived in the network with gap junctions. (C) Polar plots indicating network phase distribution for snapshots (arrows) for stimulated (colored distribution) and non-stimulated cells (gray distribution). Phase alignment of subthreshold oscillations of the inferior olivary network model was reduced in the absence of gap junctions. The lines in the middle of the plots indicate average phases for either the stimulated cells (in color) or all cells (gray). (D) Phase distribution of cells over time. Color code represents the proportion of neurons occupying a certain phase at a given time (phase bin size 2π/100, time bin is 1 ms). Dark/light bands indicate that the phase alignment propagates to most cells in the network and subsists for longer durations for the WT network. Nevertheless, contextual input overruled the phase alignment within a few hundred milliseconds (~300 ms in this case). In the absence of gap junctions, the impact of the stimulus is restricted to stimulated cells, so that the coherence induced by stimulus on the network was much smaller. (E) Same as in D but displaying averaged phase distributions for 100 stimuli. The stimulus hardly evoked an effect after 200 ms, and it did not induce any entrainment, which would have been visible as vertical bands before the stimulus. The effect of the stimulus is not stereotypical, due to dependency on network state driven by contextual input (see also Fig 9). (F) Average network synchrony triggered on the stimulus. The horizontal lines in the bottom plot are the 95% inclusion boundaries taken from the 5 s of spontaneous network behavior to contextual input (no periodic “sensory” pulses). The model network with gap junctions displayed after the early stimulus response curve an elevated synchrony plateau, which ended after about 200 ms, whereas the model without gap junctions hardly showed any secondary plateau. There was no preceding synchrony, indicating complete absence of entrainment to the periodic stimulus.
Fig 9.
Significant variability of oscillatory periods induced by contextual input on oscillation period of a single model cell.
(A) Distribution of inter-peak intervals for a single model cell under 5 s of contextual stimulus (OU current) of different amplitudes. To gauge period variability without introducing refractory periods, input levels were chosen for non-spiking subthreshold dynamics only. The unperturbed model cell (in black) produces regular periods, while weak levels of contextual input are sufficient to create substantial variability (red and blue). (B) Example traces of membrane potential and (C) phase for the single cell under perturbations, for the same random seed. Note that under contextual input there was substantial variation in periods, but similar STO periodicity despite different levels of input.
Fig 10.
No phase-dependent spiking probabilities observed in vivo.
(A) In order to test the impact of the presumed phase of the inferior olivary neurons in vivo, we applied a “gallop” stimulation pattern, alternating short (250 ms) and long (400 ms) intervals. Air puffs were delivered to the whisker pad. Complex spikes were counted in response windows (RWs) 20–200 ms post-stimulus and cells were sorted as a function of the ratio between the numbers of complex spikes in short and long intervals (indicated as horizontal dash between filled and empty bars). In this analysis, we included only the RWs that followed an RW with at least one complex spike; i.e. the RWs indicated by a dashed border were ignored. For each Purkinje cell the relative response probabilities for the long and short intervals are illustrated as the length of the filled and open bars, respectively. None of the Purkinje cells showed a significant difference in the response probability between the two intervals (all p > 0.05 on Fisher’s exact test). (B) The same for the alternation of 250 and 300 ms intervals, showing even less response bias than in A. (C) If sensory stimulation triggers complex spikes, one would expect that a higher frequency of stimulation would lead to a higher frequency of complex spikes. However, such a relation was absent. For this analysis, we included only sensory-responsive Purkinje cells of which the complex spike response exceeded a bootstrap-derived 99% probability threshold. The gray scaling of the symbols indicates the response probability. Purkinje cells displaying a higher complex spike response rate had a slightly increased firing rate upon higher stimulus frequencies, but this was far from the expected increase (based upon a linear relationship; the shaded area indicates the 25–75% range).
Fig 11.
Complex spike responses to stimuli in most Purkinje cells are not conditioned on recent history.
We quantified the impact of presumed subthreshold oscillations on the occurrence and timing of complex spikes by assuming a preferred firing frequency for each individual Purkinje cell according to peri-stimulus time histogram and auto-correlograms. This approach is illustrated by a representative Purkinje cell (A-E). A representative trace is shown in A and enlarged in B, showing the cross-stimulus interval (horizontal red line). (C-D) For each Purkinje cell, the preferred frequency was derived from the auto-correlogram (C) and the peri-stimulus time histogram (PSTH; D; cf. Fig 2). (E) Intervals between the last complex spike before and the first complex spike after stimulus for each trial yielded a model of the preferred response windows (red line). The observed probability density function was compared with a probability density function based on a uniform complex spike distribution (dotted line), an oscillatory complex spike distribution (blue line), and 9 intermediate mixed models (see Methods). (F) The distributions of the goodness-of-fit for each of the 11 models showed a clear bias towards the uniform model, casting doubt on the impact of subthreshold oscillations on sensory-induced complex spike firing. The middle line indicates the average of all runs, while the upper and lower lines indicate 75 and 25% quartiles, respectively. (G) Distributions of the goodness-of-fit of all Purkinje cells that showed clear rhythmicity (see Methods). (H) Histograms of the best mix model shown in G indicate that the impact of the subthreshold oscillations on sensory complex spike responses is present, though small.
Fig 12.
Oscillatory dynamics in spontaneous firing in vivo in the presence and absence of gap junctions.
(A) Color-coded inter-complex spike-interval (ICSI) histogram of spontaneous firing of 60 Gjd2-/- and 52 wild-type Purkinje cells, normalized to the bin with the highest complex spike count per Purkinje cell. (B) The Gdj2-/- Purkinje cell distribution shows higher Z-scores than the wild-type population, indicating a stronger rhythmicity during spontaneous firing in the absence of gap junctions (p = 0.003; Kolmogorov-Smirnov test). A Z-score >3 was taken as sign for the presence of rhythmicity–which occurred more often in the Gjd2 KO than in the WT Purkinje cells (colored fraction in the pie diagrams; p = 0.005; Fisher’s exact test). In the absence of gap junctions, rhythmic firing was more stereotypical, as illustrated by less variation in the time to peak (C; p = 0.0431; U = 1030.0; Mann-Whitney test). Note that the left panel of A is a copy of Fig 2B and presented here to facilitate comparison.
Fig 13.
Coherence of complex spike firing of Purkinje cells over time.
(A) Two Purkinje cells recorded simultaneously in a wild type (WT) mouse during spontaneous activity, showing an epoch with near-synchronous complex spike firing with both spikes in the center-time interval. (B) Another epoch of the same cell pair, but now the second Purkinje cell fired a complex spike at the next cycle, leading to a double-peaked cross-correlogram for this cell pair, seen in C. (D) Heat map representations of cross-correlograms between simultaneously recorded Purkinje cells in WT and Gjd2 KO mice. Only pairs with a significant center or side peak are plotted. The cross-correlation of each pair is normalized to the bin with maximal correlation. The latency of the strongest side peak was used to order the cell pairs (including cells where center peaks were significant, but side peaks were not). Correlation direction was selected so that the strongest side peaks were always positioned on the right side. Note that while the center peak is stronger in Gjd2 KO, the side peak is more prominent in WT pairs. This side peak can be regarded as a “network echo”. (E) Venn diagrams showing the relative occurrence of center and side peaks in the presence and absence of gap junctions. (F) In the absence of gap junctions, the center peak, but not the side peak, was stronger (left and middle panel). Likewise, the normalized ratio of the side peak vs that of the center peak was lower in the Gjd2 KO mice (right panel). * and *** indicate p < 0.05 and p < 0.001, respectively (t test).