To explore the responses of granule cells in the sublobulus C1 of the cerebellar posterior lobe to activation of the SBCs, and to calibrate our granule cell model, we made a systematic investigation, which is the subject of a separate paper [16]. The present paper includes a few examples from the data in that paper. For completeness, we give a brief description of the methods also here. The experiments were made in the acute decerebrated preparation of the cat. The cats were prepared as previously described [22]–[24]. In vivo patch clamp recordings were made from granule cells of the sublobulus C1 of the cerebellar cortex using patch pipettes pulled to 6–19 MOhm (potassium-gluconate based internal solution). In order to activate the SBC tract, the only pathway that ascends from the lumbar segments in the contralateral funiculus and terminates in sublobulus C1 [4], the contralateral lateral funiculus of the T7–T8 segments was stimulated with tungsten microelectrodes with exposed tips of 30–120 um.

The granule cells were simulated using an exponential integrate-and-fire model [25] with added stochastic component, σ_NdW described by Eq. (1). W denotes Brownian motion of the membrane potential and the entire equation is short hand notation of the corresponding stochastic integral equation (see e.g. [26] for a more complete description).(1)where V_m is the membrane potential of the model neuron. The membrane capacitance and conductance, C_m and g_L, and the reversal potential E_L where taken from [15]. The shape and properties of the action potentials are determined by the threshold V_T and the shape parameter Δ_T. V_T and Δ_T, together with the amplitude of the noise σ_N were chosen to emulate the granule cell response in vivo to single and triple pulse input. Whenever the membrane potential of the model reached 0 mV the model was considered to have produced an action potential, and the membrane potential was reset. The synaptic input I_SBC to the granule cell models was modeled as the AMPA-receptor synapses in [27] in Eq. (2).(2)where s_i is the activation level of the i^th SBC to granule cell synapse that is set to 1 whenever the i^th SBC fires. τ_decay is the time constant that determines how fast s returns to 0 following a spike and E_AMPA is the reversal potential for the post-synaptic current. The synaptic conductances, g_i,SBC were chosen to create unitary EPSPs of the same amplitude as the EPSPs in [15] (from Fig. 6 in that paper). The parameters used to simulate the granule cells are shown in Table 1.

The spike trains of the SBCs where created using an inhomogeneous point process, with an instantaneous rate related to the membrane potential of the neuron [28]. The granule cells and spinal neurons were simulated using different models of spike generation for two reasons. Firstly, VSCT neurons, of which the SBCs are a subpopulation, work as straight-forward rate coders during fictive locomotion [9] whereas granule cells differ from these and other rate coding neurons by having a great gap between the resting potential and firing threshold [15] allowing them to have fast transient responses that cannot be captured by the escape rate model but can be described by the integrate-and-fire model. Secondly, whereas the data from VSCT/SBC neurons exist in the form of firing rate modulations to relatively slowly modulated intracellular depolarizations, suitable for escape rate models such as the one we use [28], the data available for granule cells responding to SBC tract stimulation, which generates artificially synchronized and very sharp population EPSPs, is more suitable to approximate with the integrate-and-fire model.

The applicability of the used inhomogeneous point process, and its verification in in vivo recordings from spinal neurons, is treated in detail in a separate paper [28]. Briefly, the spike firing rate was described by a modulated log-normal hazard function that was fitted to respond with the same intensity as the VSCT neurons in [9], [10] when subjected to input that depolarizes the membrane potential in the range of 0 and 7 mV from the resting potential, similar to the maximal depolarization of 7.1 mV recorded in VSCT neurons under fictive locomotion [9].

The modulation of the intensity of the excitatory RST and inhibitory Ib input to the RST was interpolated from [19] using cubic splines to recreate the overall shape of the RST intensity and EMG activity of Srt and VL.

The lognormal distributions were modulated according to Eq. (3) [28].(3)where E_ISI is the mean and Std_ISI is the standard deviation of the lognormal distribution of the inter-spike intervals (ISI). x is a dimensionless parameter that translates between the input to the model and the parametric relationship between E_ISI and Std_ISI. c_x and Δ_x are parameters that govern the parametric relationship between the mean and standard deviation of the model, and c_I and Δ_I are parameters that govern the relationship between the input current and the modulation of the firing statistics. The synaptic input to the SBC model, I_RST+I_Ib, were modeled as a current created from a linear combination of excitatory input from RST and inhibitory input from inhibitory interneurons. The parameters that were used to simulate the SBC model comes from one of the spinal interneurons in [28] and can be found in Table 1.

The mean and standard deviation of a lognormal distribution can be converted to the mean σ and standard deviation μ of the underlying normal distribution according to Eq. (4) as these are commonly used to parameterize the log-normal distribution.(4)Eq. (5) is the hazard rate of the lognormal distribution that was used to simulate the inhomogeneous point process.(5)where f(t) is the probability density function and F(t) the cumulative density function of the log-normal distribution and h(t) is the instantaneous hazard rate, determining the probability of a spike at time t. Φ denotes the cumulative density function of the standard normal distribution.

The model equations were solved using a standard 4^th order Runge-Kutta solver with a Δt = 0.1 ms timestep using custom designed software and the subsequent data analysis was performed with MATLAB (Mathworks). The same time step was used to determine whether the SBC models fired an action potential using the hazard rate from Eq. (5) multiplied with the time step, h(t)Δt, as the probability of an action potential being generated.

The instantaneous firing frequency (IFF) was used as a measure of the spike response in order to facilitate comparison of the results. The time of individual spikes were binned and the IFF was calculated as the probability of a spike during that bin divided by the bin width. In cases where the firing probability is more applicable (i.e. during transient behavior) it is displayed alongside the IFF.

In order to evaluate whether a recorded behavior was predicted by the model, the 95% confidence interval of the IFF histograms of the model were created using a Monte Carlo simulation. The model was simulated to respond to the same number of traces as was recorded in vivo. This procedure was repeated 500 times which allowed the mean response and confidence interval to be calculated for each bin. If the in vivo response fell within the confidence bounds in more than 95% of the bins it was considered to predict the in vivo data. Note that long-latency responses were not included in the analysis.

The membrane properties and the mossy fiber synaptic properties of our granule cell model were based on the data presented in a previous investigation of granule cells in the anterior lobe using the in vivo whole cell patch clamp technique [15]. However, in order to verify that the granule cells in the region that receives SBC tract inputs behave in a similar fashion, we aimed to also make whole cell recordings from these cells in the sublobulus C1. Figure 2 presents one such recording and the relationship between the injected current and average firing frequency in this cell. We took this example as a verification that our granule cell model, based on recordings from granule cells primarily in the paravermis of the anterior lobe [15], also applied to the sublobulus C1 of the paravermis of the posterior lobe.

In order to calibrate the spike firing properties of our granule cell model, we used data obtained from recordings of sublobule C1 granule cell responses to SBC tract single and triple pulse stimulation [16] (Fig. 3). In these cases, the intensity of the response was comparable to those evoked through the cuneocerebellar and reticulocerebellar tracts in granule cells recorded in the anterior lobe, for which a more systematic analysis of the composition of the synaptic input has been made previously. At least for the anterior lobe, this previous analysis indicated that all four mossy fiber inputs to the granule cell were derived from the same input source [15], [18]. Therefore, in the present paper, we simulated only granule cells that receive SBC mossy fiber input on all four dendrites. For a granule cell in which a lower proportion of the mossy fiber synaptic inputs were driven by the SBC tract input, the synaptic input signal would display a similar profile as we present in the remainder of this paper, but the amplitude of the modulation of the membrane potential and the influence on the spike output would be lower.

Under these modeling conditions, we adapted the spike generation of two granule cells, chosen because of their different patterns of spike responses to the SBC tract stimulation [16], so that the model displayed comparable spike responses as in the in vivo setting. The model parameters where chosen such that the model could predict the in vivo responses within a 95% confidence bound (Fig. 3). Even though the in vivo responses to SBC tract stimulation of the two cells were distinctly different, the model could capture both response patterns solely by changing the latencies of the four individual mossy fiber synaptic inputs and the resting potential of the granule cells (−58 mV and −57 mV respectively). The fast response of the second cell (Fig. 3 C,D) could only be captured using a synchronous synaptic input from four SBCs (left inset Fig. 3C), while the more slow response of the first cell required different response latencies for the four SBC synaptic inputs (Fig. 3A,B). The scenario of different response latency times of SBC inputs is compatible with the known differences in the conduction velocities of VSCT neurons [30]. Note that later parts of the recorded granule cell responses (see for example Fig. 3A) is likely to represent input derived from long-latency synaptic activation of the SBCs via other pathways (i.e. an artifact of the experimental setup) [16] and we did not aim to capture this behavior in our model.

Since the lumbal spinocerebellar systems and the reticulospinal tract are strongly modulated under locomotion [10], [31], [32], as are the neurons of the paravermal cerebellar cortex [33]–[37], we wanted to design our model to explore the spinocerebellar interactions in the cat during this behavioural context. Therefore, the SBCs were simulated to receive excitatory drive from the RST and inhibitory input from spinal interneurons, which were simulated to receive only primary afferent input of the Ib muscle afferent type.

SBCs are innervated by a massive (approximately 5000) number of excitatory synaptic inputs from the RST and a comparable number of synaptic inputs from inhibitory interneurons [7], [8]. At the population level, these synaptic inputs can generate a total depolarization of up to 5 mV and a total hyperpolarization of about 7 mV when activated synchronously, using electrical stimulation of the descending tracts or afferent nerves [7], [8], [11], [38], [39]. It follows, similar to the magnitude of synaptic weights in other synaptic junctions of the cat spinal cord [40], that the responses of the individual synaptic connections are likely to amount to only a few µVs in amplitude in SBCs. On the other hand, since the SBCs receive so many RST synapses and inhibitory synapses from interneurons activated by Ib afferents, concerted activation of these inputs will be translated to relatively massive excitatory and inhibitory modulations of the intracellular potential (shown for VSCT neurons under fictive locomotion [9]), which are much slower than the responses obtained by synchronous fiber activation obtained on tract stimulation and nerve shock stimulation [7], [8]. The amplitude of these intracellular membrane potential changes should be related to the average firing rate of the neurons that converge onto the individual SBC. In our simulation of the intracellular signals of the SBCs, these smooth total synaptic input signals are reduced to a single input current that correspond to the average, summed activity of the populations of excitatory and inhibitory neurons that innervate the SBC.

The time-course of the RST excitatory modulation was adapted from [19], which illustrate three examples of RST neuron modulation during locomotion (Fig. 4A). We adapted functions to reproduce the individual time-courses of RST neuron activation (see Methods) and then summed these functions to generate an average RST neuron activity used to drive the excitatory input to the simulated SBCs. The simulated Ib afferent activity was similarly taken directly from EMG recordings of the specific muscles of interest obtained during locomotion [19]. This was motivated by a study that has shown that during locomotion the modulation of an ensemble of Ib afferents essentially mirrors the EMG activity of the corresponding muscle [21]. An essential assumption of the present model is that the Ib afferent activity is transmitted linearly via the spinal interneurons to the SBCs – consequently, the simulated Ib afferent activity was simply provided with a negative synaptic weight, with variable gain as shown below, to the SBCs. We focused on Ib afferent input from the vastus lateralis (quadriceps) muscle and the sartorius muscle (Fig. 4B), whose Ib afferents were given different relative weights in the simulated SBCs to explore the effects of varying the convergence pattern in the SBC/spinal interneurons.

In order to validate that the modelled SBCs could display similar input-output relationships as actual SBCs, we provided our SBC model with the intracellular signal recorded from a VSCT neuron during a step-cycle and compared the spike output with that recorded from the same neuron (recorded data taken from figure panels in Ref. [9]) (Fig. 5). The model captured both the initial steep rise in firing frequency and its slow decay following the initial peak (Fig. 5). 200 ms into the step, the reproduction was poorer, but the discrepancy can be explained by that the recorded intracellular signals differed from step to step and the recorded firing displayed was an average of 12 step cycles.

The result of the SBC simulation can be seen in Figure 6. The amplitude of the simulated intracellular signal (7.0 mV), displayed without action potentials (Fig. 6A), corresponded to that recorded in VSCT neurons during fictive locomotion (7.1 mV) [9], i.e. without Ib inhibition as the muscle nerves were cut. It generated a spike output (with a peak amplitude of 264 Hz) in the modeled SBC neuron which was comparable to that recorded in VSCT neurons during actual and fictive locomotion (peak amplitude of 100–300 Hz) [9], [10]. Figure 6B illustrates the average instantaneous SBC firing rate across 1000 step cycles. When the simulated SBC neurons were provided with inhibitory input driven by simulated Ib afferent activity, the intracellular signal and the firing rate were modulated accordingly (Fig. 6C,D).

Figure 7 illustrates the intracellular membrane potential and spike responses of the two simulated granule cells that each received inputs from four simulated SBCs during one locomotion step cycle. Fig. 6A illustrates the simulated response of the granule cell simulated in Fig. 3A,B (cell #1) whereas Fig. 6B illustrates the granule cell simulated in Fig. 3C,D (cell #2). In the model, the granule cell simply integrates the EPSPs evoked by spikes in the individual SBCs (Fig. 6B) and the Ib inhibition in the SBCs (Fig. 6D) is expressed as a reduction in the number of EPSPs generated in the granule cell.

We next used the modeled network to explore the effect of varying the synaptic weights with which the Ib inhibitory inputs modulated the SBC neurons. Figure 8 illustrates how the granule cell activity varied across different levels of VL and Sart Ib afferent inhibitory synaptic weights on SBCs. As the inputs were linearly combined in the SBCs, increasing inhibitory synaptic weights resulted in linear reductions in the granule cell spiking activity corresponding to the specific temporal profiles of activation of the respective muscle. We made this simulation both for granule cell #1 (Fig. 7A,B) and cell #2 (Fig. 7C,D). Note that the two granule cells, with different responses to SBC tract stimulation (see Fig. 3) still fired with a similar temporal topography and overall firing frequency.

The main potential limitations of the present study lie in the generation of spike output in the SBCs and in its afferent inhibitory interneurons. However, spinal interneurons and spinocerebellar neurons appear to be essentially linear integrators [28], [41], [44]–[49] and the details of how the spikes are generated may matter little in a system that seems to operate primarily according to the rate coding principle. The convergence pattern of Ib afferents from different muscles in the inhibitory interneurons are more extensive than in our simulation [11] but the effects of possible additional inhibitory synaptic inputs from other muscles are illustrated in principle in Fig. 7. Basically, adding inhibitory effects from other muscles is equal to adjusting their synaptic weights, and similar to the effects of varying the amount of inhibition in Fig. 7 the effect of other muscles will be linearly added.

Under the conditions tested here, the granule cells responded to a linear combination of the Ib afferent sensory input and the RST motor command. While such an organization is likely for the descending RST signals that directly excite the SBCs, the sensory inhibition is mediated through at least one additional synaptic relay, i.e. the Ib inhibitory interneuron, which may conceal additional complexity. These interneurons may not be limited to mediating Ib input, but may in many cases mediate a combination of input from group Ia, group Ib and group II muscle afferents [1], [11], [12], the descending motor command [40], and also other local interneurons. Hence, the SBCs would via these interneurons be able to sample the state of the spinal cord and thereby providing a higher level of representation of the local sensorimotor information to the cerebellum [43]. The additional functions that could be represented in the system due to any complexity in how the Ib interneurons integrate information is beyond the scope of the present paper since it would in that case be a property of the interneurons. But these functions would be expected to be linearly combined with the functions we explored in the present paper, due to the linearity of spinal interneurons [16].

The spinal neuronal network described here illustrates the formation of individual projections through a multi-dimensional sensorimotor space, the theoretical advantages of which were explored in [3] and for which there is direct experimental support [41]. In the present paper, efferent signals from the RST, which likely represent a set of motor dimensions, were recombined with Ib afferent signals from a pair of muscles (each representing a single sensor dimension). By varying the strength of the Ib inhibition from one of the muscles, the direction of the projection in these input dimensions will change. The synaptic weights by which the Ib afferents actually influence the SBC neuron presumably represent a learnt pattern, which is particularly useful to the spinocerebellar system [3]. Although we do not presently know how this learning would work at the level of the spinal cord (such learning has been demonstrated during development for cutaneous inputs to spinal neurons and withdrawal reflexes [50], [51]), in the present study we explored a range of scenarios for how different weights in this particular synaptic junction would influence the spinocerebellar processing (Fig. 7). If the signals that converge in the SBCs are functionally related, which could emerge as a result of developmental learning and is a main assumption of the present model, it offers the advantage of supplying the granule cells with relevant recombined signals [3] rather than a complete random arrangement, where each granule cell would sample a random set of input signals (i.e. the scenario portrayed in the popular models of granule layer processing originating from Marr [52]). Without this functionally selective convergence upon the granule cells, there would be a large number of granule cells that received functionally disparate input that would have no functionally relevant, non-linear interactions during normal behavior and would therefore be of limited use to the cerebellum [3].

After the information is generated at the level of the granule cells, it is transmitted to the output cells of the cerebellar cortex, i.e. the Purkinje cells as well as the inhibitory interneurons innervating the Purkinje cells. Here, the information can be integrated with other sources of information, such as sensory input from the dorsal column nuclei [15], [18] and neocortical input via the pontine nuclei. How the information in this integration is weighted is probably to a large extent determined by plasticity processes, which allows both Purkinje cells and interneurons to pick out and weigh the specific pieces of information that is required for fulfilling their task of regulating the cerebellar output [3], [53], [54]. How the divergent deep cerebellar nuclear output [55], [56] subsequently is integrated with and contributing to the synergy control of the motor systems of the brain [42] remains one of the greatest challenges to the field of motor control neuroscience.