Gradients in the cerebellar cortex enable Fourier-like transformation and improve storing capacity

Cerebellar granule cells (GCs) making up majority of all the neurons in the vertebrate brain, but heterogeneities among GCs and potential functional consequences are poorly understood. Here, we identified unexpected gradients in the biophysical properties of GCs. GCs closer to the white matter (inner-zone GCs) had higher firing thresholds and could sustain firing with larger current inputs. Dynamic clamp experiments showed that inner- and outer-zone GCs preferentially respond to high- and low-frequency mossy fiber inputs, respectively, enabling to disperse the mossy fiber input into its frequency components as performed by a Fourier transformation. Furthermore, inner-zone GCs have faster axonal conduction velocity and elicit faster synaptic potentials in Purkinje cells. Neuronal network modeling revealed that these gradients improve spike-timing precision of Purkinje cells and decrease the number of GCs required to learn spike-sequences. Thus, our study uncovers biophysical gradients in the cerebellar cortex enabling a Fourier-like transformation of mossy fiber inputs.


Introduction 35
Digital audio compression (e.g., 'MP3'; Jayant et al., 1993) and image 36 compression (e.g., 'JPEG'; Wallace, 1992) rely on Fourier transformations, which 37 decompose a signal (e.g., sound amplitude as a function of time or image 38 intensity as a function of space) into its frequency components (power as a 39 function of frequency). By storing these frequency components with different 40 precision depending on psychophysical demands of hearing and seeing, the 41 overall storage capacity can be increased dramatically. between lobule V and IX differed (Figure 1-figure supplement 1), consistent 125 with previously described differences in, e.g., the firing frequency in vivo between 126 these two lobules ( tested whether these gradients were also present at a later developmental stage. 137 Recordings obtained from GCs in lobule V in animals between 80 and 100 days 138 of age showed the very similar gradients as in young animals (Figure 1-figure  139 supplement 2). Together, these data show a prominent gradient in the 140 electrophysiological properties of GCs over the depth of the granule cell layer, 141 and that this gradient can consistently be found in different lobules and ages. 142

Voltage-gated potassium currents are larger at inner-zone GCs 143
To investigate possible causes for the gradient in the biophysical properties, we 144 investigated voltage-gated potassium (Kv) currents by performing voltage-clamp 145 recordings in outside-out patches from somata of inner-and outer-zone GCs in 146 lobule V (Figure 2A). The maximum Kv current was significantly higher in inner-147 zone GCs (282 ± 29 pA, n = 48) compared with outer-zone GCs (221 ± 28 pA, n 148 = 54, P Mann-Whitney = 0.02; Figure 2B). Neither the steady-state activation curve 149 ( Figure 2C) nor the degree of inactivation ( Figure 2D) was different between the 150 two GC populations. Furthermore, steady-state inactivation, which was 151 investigated with different holding potentials, was similar between inner-and 152 outer-zone GCs (Figure 2-figure supplement 1). These data suggest that 153 inner-and outer-zone GCs have a similar composition of Kv channels, but inner-154 zone GCs have a higher Kv channel density. The here observed larger Kv 155 currents in inner-zone GCs are consistent with the short action potential duration 156 of inner-zone GCs (cf. Figure 1). Thus, our data provide a biophysical 157 explanation for the observed gradient in GC properties. 158

MF inputs are differentially processed by inner and outer GCs 159
The gradient within the GC layer creates an optimal range of input strengths for 160 each GC. To test how this gradient impacts the processing of synaptic MF inputs, 161 we performed dynamic clamp experiments to test whether different MF input 162 frequencies differentially affect spiking in inner-and outer-zone GCs ( Figure 3A). 163 We first recorded excitatory postsynaptic currents (EPSC) from GCs located at 164 inner-or outer-zone of lobule V after single MF stimulation. We found no 165 significant differences in the amplitude nor in the kinetics of EPSCs in inner-and 166 outer-zone GCs (Figure 3-figure supplement 1). 167 Individual MFs span the entire depth of the GC layer, contacting both inner-and 168 outer-zone GCs (Krieger et al., 1985;Palay and Chan-Palay, 1974).  Silver et al., 1992). Therefore, we could use the dynamic clamp technique 172 to implement the conductance of identical MF signals in inner-and outer-zone 173 GCs based on the measured EPSC kinetics. We first applied input of a single MF 174 with Poisson-distributed firing-frequencies ranging between 30 and 500 Hz for 175 300 ms duration while changing the resting membrane potential to simulate the 176 large variability of membrane potential of GCs observed in vivo (Chadderton et 177 al., 2004). In line with the observed gradients in the electrophysiological 178 properties of GCs, inner-zone GCs fired fewer action potentials compared with 179 outer-zone GCs in response to low-frequency MF inputs at a membrane potential 180 of ~-90 mV (Figure 3B,C). In contrast, inner-zone GCs fired more action 181 potentials compared with outer-zone GCs in response to high-frequency MF 182 inputs at a membrane potential of ~-70 mV. In vivo, such a depolarization would 183 be caused by less inhibition and/or additional MF inputs. These data suggest that 184 In addition to the above results obtained from lobule V, similar gradients in both 244 axon diameter and axon conduction speed were found in lobule IX (Figure 6-245 figure supplement 1). This suggests that gradients in axon diameter and axon 246 conduction speed are general features of the cerebellar cortex. 247 A possible confounder of our results could be an overrepresentation of large-248 diameter Lugaro cell axons within inner-zone PFs (Dieudonne and Dumoulin, 249 2000). However, this would predict that the histogram of the axon diameters 250 shows two distinct peaks with varying amplitude. Instead, we observed a single 251 bell-shaped distribution in each PF zone (Figure 6-figure supplement 2), 252 arguing that the measured differences between axon diameters were not due to 253 varying contributions from Lugaro cell axons, but reflect the differences between 254 inner-, middle-and outer-zone PFs. 255

PCs process inner-, middle-, and outer-zone PF inputs differentially 256
Our data thus far indicate that GCs and PFs are adapted to different MF input 257 frequencies and conduction velocity, respectively. This arrangement could in 258 principle provide PFs with functionally segregated information streams that are 259 differentially processed in PCs. To investigate this possibility, we made whole-260 cell current-clamp recordings from PCs in sagittal slices of the cerebellar vermis. 261 PCs were held at a hyperpolarized voltage to prevent spiking and to isolate 262 excitatory inputs. Electrical stimulation of the PFs was alternated between inner-, 263 middle-and outer-zones and adjusted to obtain similar amplitude EPSPs in all 264 zones ( Figure 7A,B). Stimulation of inner-zone PFs resulted in EPSPs (Barbour, 265 1993; Roth and Häusser, 2001) with shorter rise and decay times compared with 266 EPSPs obtained from stimulating outer-zone PFs (rise 20-80 : inner: 0.57 ± 0.04 ms, 267 n = 12; middle: 0.93 ± 0.17 ms, n = 4; outer: 1.83 ± 0.33 ms, n = 12 (P ANOVA = 268 0.009; decay: inner: 21.9 ± 1.5 ms, middle: 39.7 ± 1.1 ms outer: 40.8 ± 4.1 ms; 269 P ANOVA = 0.0004, Figure 7C). These results suggest that inner-zone PF inputs 270 undergo less dendritic filtering in PCs compared with outer-zone PF inputs (De 271 Schutter and Bower, 1994a, b;Roth and Häusser, 2001) but see (De Schutter  272 and Bower, 1994c). To investigate high-frequency inputs to PCs, we elicited five 273 EPSPs at 100 Hz and 500 Hz (Figure 7D,E). Individual EPSPs evoked from 274 inner-zone PFs showed clear individual rising phases and peaks between each 275 stimulus and less summation compared with outer-zone PFs (Figure 7D-F). 276 These results suggest that inner-zone PFs can transmit timing information more 277 faithfully compared with outer-zone PFs and thus control spike timing of PCs 278 more precisely. 279 The observed neuronal gradients increase storing capacity and improve 280 temporal precision of PC spiking 281 Thus far we have described a prominent gradient in the electrophysiological 282 properties of GCs over the depth of the GC layer that enables inner-and outer-283 zone GCs to preferentially respond to high-and low-frequency inputs, 284 respectively. The different frequency components are transferred via specialized 285 PFs, which enable PCs to interpret high-frequency signals rapidly at the base of 286 their dendritic trees and low-frequency signals slowly at more distal parts of their 287 dendritic trees ( Figure 8A). or bursting (Rancz et al., 2007) in vivo-like spiking sequences. By changing the 294 synaptic weights of the GC to PC synapses, the PC had to acquire a target 295 spiking sequence with regular 80-, 40-and 120-Hz firing ( Figure 8B). The 296 algorithm for changing the synaptic weights was a combination of a learning 297 algorithm based on climbing-fiber-like punishments and an unbiased 298 minimization algorithm (see Methods). 299 We first compared a model without gradients, where the parameters were set at 300 the average of the experimentally determined values, with a model including all 301 experimentally determined gradients (black and red, respectively, throughout 302  400 GCs with all gradients, the model without gradients required 800 GCs (cf. 315 red arrows in Figure 8C). This indicates that for a cerebellum exploiting gradients 316 in the GC layer, the number of GCs can at least be halved to obtain a certain 317 temporal precision compared with a cerebellum containing no gradients. 318 To investigate the relative contribution of each of the gradients, we tested models 319 containing each gradient in isolation, resulting in intermediate van Rossum errors 320 (blue, yellow, and green in Figure 8C  To further investigate the interplay of the different gradients, we investigated a 325 model containing all gradients, but the connectivity between GCs, PF action 326 potential speed, and PC EPSP kinetics were randomly intermixed (red dashed 327 lines in Figure 8C-E). The network benefits from these intermixed gradients, but 328 maximum optimization can only be obtained with correct connectivity (   The time constant of the van Rossum error can be decreased or increased to 331 investigate spike timing or slower changes in firing rate, respectively. The impact 332 of the gradients increased with increasing time constant (Figure 8-figure  333 supplement 1A,B), indicating that rate coded signaling especially benefits from 334 the here described gradients. To specifically test the effect of gradients on the 335 cerebellum's ability to switch between firing frequencies, we made sigmoid fits 336 around the times of firing rate changes. The transition time (t T ; see methods) 337 from these fits showed that models with all gradients showed on average 20% 338 faster switching between firing frequencies than models without any gradients 339 In this study, we describe a gradient in the biophysical properties of superficial to 357 deep GCs, which enables the GC layer to perform a Fourier-like transformation 358 of the MF input. Furthermore, we show that the downstream pathways from GCs 359 to PCs are specialized for transmitting the frequency band for which the 360 corresponding GCs are tuned to. Finally, computational modeling demonstrates 361 that both the gradients in the GC layer and the specialized downstream pathways 362 improve the spiking precision, accelerate the change of firing frequency of PCs, 363 and increase storing capacity in the cerebellar cortex. 364

Fourier-like transformation in the cerebellar cortex 365
Our data demonstrate that outer-zone GCs preferentially fire during MF input with 366 low frequency ('low-frequency' GCs, magenta in Figure 9A), whereas inner-zone 367 GCs preferentially fire during MF input with high frequency ('high-frequency' 368 GCs, green in Figure 9A). The separation of a signal into its frequency 369 components resembles a Fourier transformation ( Figure 9B). The analogy with a 370 Fourier transformation has the limitations that (1) a single MF cannot transmit two 371 frequencies simultaneously but only separated in time (cf. example in Figure 9A) 372 and (2) concurrent inputs from two MFs with different frequencies synapsing onto 373 a single GC cannot be separated. Yet, our data indicate that the entire GC layer 374 with several MFs sending various frequencies to numerous GCs can execute a 375 Fourier-like transformation. In analogy to the dispersion of white light in an optical 376 prism into its spectral components, the broadband MF signal is separated into its 377 spectral components with inner-to outer-zone GCs preferentially transmitting the 378 high-to low frequency components, respectively. Such a separation offers the 379 chance to process each frequency component differentially. In addition to these two known axes of heterogeneity, we described an axis that 415 is orthogonal to the surface of the cerebellar cortex. This 'depth' axis causes 416 inner-zone GCs to be tuned to higher frequencies than outer-zone GCs. This interesting to elucidate in, as much previous models, consisting of more uniform 476 MF inputs, would benefit from the here-observed biophysical gradients. 477 To implement these gradients in a model we used a simplified cerebellar circuitry 478 that does not consider active dendrites (Llinás and Sugimori, 1980)  Therefore, the dramatic increase in storing capacity for precise PC spiking 499 provides an evolutionary explanation for the emergence of gradients in the 500 neuronal properties. 501

Functional implications for other neural networks 502
Based on the described advantages of the Fourier transformation for rapid and 503 storing-efficient information processing, we hypothesize that other neural 504 networks also perform Fourier-like transformations and use segregated 505 frequency-specific signaling pathways. To our knowledge this has rarely been 506 shown explicitly, but similar mechanisms might operate, for example, in the 507 spinal cord network: descending motor commands from the pyramidal tract send 508 broadband signals to motoneurons with different input resistances resulting from 509 differences in size. This enables small motoneurons to fire during low-frequency 510 inputs and large motoneurons only during high-frequency inputs (Henneman et 511 al., 1965). Furthermore, specialized efferent down-stream signaling pathways 512 innervate specific types of muscles with specialized short-term plasticity of the 513 corresponding neuromuscular junctions (Wang and Brehm, 2017). GCs were implemented as integrate and fire models with the following 750 parameters: the membrane resistance was linearly varied between 450 MW for 751 inner GCs to 800 MW for outer GCs ( Figure 1H) and the threshold was linearly 752 varied between -37 mV for inner GCs to -42 mV for outer GCs ( Figure 1G) (Figure 8-figure supplement  805   1A,B). The van Rossum error was defined as the integral of the square of the 806 difference between these two convolved traces. We also tested another 807 algorithm to calculate the van  Bar graphs represent the firing threshold of GCs from inner (dark-931 green), middle (dark-grey) and outer-zone (dark-magenta). The light-932 colored bar graphs in the background are the data from lobule V 933 shown in Figure 1. Firing threshold is higher in inner-compared to 934 outer-zone GCs from lobule IX, and with the same current injection, 935 GCs from lobule IX fire action potentials faster compared to lobule V. 936 The numbers of recorded GCs for lobule IX (n) are indicated (P Tukey = 937 0.007 for inner-vs outer-zone GCs). 938 C.
Average current needed to elicit the maximum number of action 939 potentials for of inner-(green), middle-(grey) and outer-zone GCs 940 (magenta) (P Tukey = 0.002 for inner-vs outer-zone GCs Since the mean current threshold is higher compared to young 994 animals only 8 out of 21 GCs from inner-and 10 out of 22 GCs from 995 outer-zone already fired action potentials at a current injection 996 H. Delay of the first action potential after a current injection of 60 pA from 997 inner-, middle-and outer-zone GCs from lobule IX compared to lobule 998 V. 999 I. The action potential half-duration of inner-zone (dark-green) GCs from 1000 old animals is shorter compared with outer-zone (dark-magenta) GCs. 1001 1002 Histograms of the diameter of inner-, middle-and outer-zone axons in the 1131 molecular layer, indicating that the change in axon diameter is not due to an 1132 increased fraction of larger diameter axons from non-GC cells (e.g. Lugaro cells). 1133 Instead the entire distribution of the axon diameters is shifted between zones. 1134 Data were fit with a skewed Gaussian function: where a is the amplitude, d the diameter, and d0 the diameter at the peak. ds 1136 and b represent parameters related to the width and the skewness, respectively. 1137 The peak is indicated by a vertical line. Illustration of a broadband MF inputs conveying a sequence of low, 1217 high, and low firing frequency. Inner-zone GCs will preferentially fire 1218 during high-frequency inputs ('high-frequency' GC) and outer-zone 1219 GCs during low-frequency inputs ('low-frequency' GC). 1220 B. Schematic illustration of the signal flow through the cerebellar cortex. 1221 The Fourier-like transformation in the GC layer is illustrated as an 1222 optical prism separating the spectral components on the MF input. 1223 Thereby the MF signal in the time domain is transformed to the 1224 frequency domain and sent via specialized signaling pathways in the 1225 molecular layer to the PC. 1226