# Coding Capacity of Purkinje Cells with Different Schemes of Morphological
Reduction

Lingling An†, Yuanhong Tang†, Quan Wang†, Qingqi Pei†, Ran Wei†, Huiyuan
Duan†, Jian K. Liu‡
†School of Computer Science and Technology, Xidian University, Xi’an, China
‡Centre for Systems Neuroscience, Department of Neuroscience, Psychology and
Behaviour, University of Leicester, Leicester, UK
Correspondence<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The brain as a neuronal system has very complex structure with large diversity
of neuronal types. The most basic complexity is seen from the structure of
neuronal morphology, which usually has a complex tree-like structure with
dendritic spines distributed in branches. For simulating a large-scale network
with spiking neurons, the simple point neuron, such as integrate-and-fire
neuron, is often used. However, recent experimental evidence suggests that the
computational ability of a single neuron is largely enhanced by its
morphological structure, in particular, by various types of dendritic
dynamics. As morphology reduction of detailed biophysical models is one of
classic questions for systems neuroscience, much effort has been taken to
simulate a neuron with a few compartments to include the interaction between
soma and dendritic spines. Yet, novel reduction methods are still needed to
deal with complex dendritic tree. Here by using ten individual Purkinje cells
of the cerebellum from three species of guinea-pig, mouse and rat, we consider
four types of reduction methods and study their effects on the coding capacity
of Purkinje cells in terms of firing rate, timing coding, spiking pattern, and
modulated firing under different stimulation protocols. We find that there is
a variation of reduction performance depending on individual cells and
species, however, all reduction methods can preserve, to some degree, firing
activity of the full model of Purkinje cell. Therefore, when stimulating
large-scale network of neurons, one has to choose a proper type of reduced
neuronal model depending on the questions addressed. Among these reduction
schemes, Branch method, that preserves the geometrical volume of neurons, can
achieve the best balance among different performance measures of accuracy,
simplification, and computational efficiency, and reproduce various phenomena
shown in the full morphology model of Purkinje cells. Taken together, these
results suggest that Branch reduction scheme seems to provide a general
guideline for reducing complex morphology into a few compartments without loss
of basic characteristics of firing property of neurons.

##  I Introduction

A single neuron is thought as the basic computation unit in the complex
neuronal system. Understanding the computations and dynamics of single neurons
is an integral part of explaining their functional roles within a large neural
network. However, there is a complex dendritic structure for a single neuron’s
morphology. The question is what are the key properties of neuronal morphology
for understanding the complex process such as information processing and
dynamic behavior emerged from a network of neurons. How one can reduce the
whole morphology but still keeping these essential dynamics? To address these
questions, there are a number of methods and models developed to describe
firing properties of single neurons at different levels of abstractions and
reductions. A simple, perhaps the most compact version of a neuronal model is
the so-called integrate-and-fire (IAF) model [1]. The IAF model of neuronal
dynamics is considered as a simple way to explain certain aspects of neuronal
behaviors [2, 3]. In particular, it has been used mostly for simulation of
large-scale neuronal networks. However, with the advancement of experimental
techniques for characterizing fine dynamics of neurons, there is an increasing
volume of evidence that the IAF approach cannot capture the dynamics and
computations of real neurons [4, 5, 6, 7, 8], in which neuronal morphology,
ion channels, and synapse distributions, all affect the activity of neurons
[9]. The classical Rall cable theory is useful to understand the contribution
of neuronal spatial structure to its function and dynamics [10]. By extending
the Rall cable theory, the real neuronal morphology can be reconstructed based
on the anatomy to ensure that its electrophysiological characteristics are
unchanged. Although the detailed model can describe the dynamics of individual
neurons very well, their high dimensionality and complex spatial structure
make the calculation expensive and unsuitable for large-scale network
simulation.

The classical Hodgkin-Huxley single-compartment model ignores the spatial
structure of neurons to mainly explain the ionic mechanisms and how action
potentials are initiated and propagated in neurons [11]. However, the recent
theoretical work shows that the single somatic point neuron is not enough to
capture the detailed firing activities of Purkinje cell observed in
experiments [7]. The neuronal model with at least two compartments of soma and
dendrite is necessary. Similarly, the reduced model with only one or a few
dendritic compartments is usually sufficient to understand detailed neuronal
activities [12, 13, 8, 14]. For large-scale network studies, reduced models
provide a good balance between biological activity and computational
efficiency [9]. However, what kind of reduction schemes for simplifying the
whole dendritic morphology is still less well understood [8, 14].

Here we address this question by focusing on the Purkinje cells (PCs) of the
cerebellum. The cerebellum, as one of the most well-studied brain areas,
traditionally plays an essential role for motor control [15, 16, 17, 18] and
learning of vestibular-ocular reflex [19, 20, 21] and eyelid reflex regulation
[22, 23]. In recent years, a large number of studies have shown that the
cerebellum is also involved in the processing of information such as
cognition, language, attention, and memory [24, 25, 26, 27, 28, 29, 30].

PCs, as the only output neurons in the cerebellum, is an indispensable
component in the mechanism of synaptic plasticity in cerebellar learning. PCs
receive parallel fiber (PF) input that generates high-frequency simple spikes
(SSs) to predict ongoing movements [31]. The high-frequency SS discharge of
PCs encodes the information about movements, including performance errors and
kinematics [32, 33]. Importantly, SS modulation both leads and lags behavior,
which means that individual PCs may carry predictive and feedback signals
about motor commands and corresponding behaviors [34, 35, 36, 37]. PC has a
tree-like morphology structure with an intricately elaborate dendritic arbor
which can be characterized by a large number of dendritic spines distributed
in its dendritic branches. Given such a sophisticated dendritic structure, it
is necessary to find a suitable level of abstraction with a reduced
morphological structure to understand the working mechanisms of PC for
integration of PF synaptic inputs.

In a previous study [14], a reduction method based on Strahler analysis of
neuron morphology was applied to PC with arbitrary dendritic distribution of
ion channels and synaptic inputs and without any fitting or tuning procedures.
In particular, this reduction method can ensure somatic membrane potential
trace accurate while reducing the runtime of simulation significantly. In this
study, we systematically model the morphology of PC from three species of
guinea-pig, mouse and rat, and investigate the effect of four different
reduction schemes on the coding capacity of PC, in terms of firing rate, spike
timing, spike pattern, and modulation of firing amplitude and phase, with
several types of stimulation protocols of PF synaptic inputs, including
Poisson and modulated renewal process.

We find that the PC firing rate coding, which is the input-output relationship
of the PC firing activity under different frequencies, is slightly different
in low frequency stimuli but significantly different in high frequency stimuli
at a millimeter-scale reaction under different reduction schemes of
morphology. The PC spike timing coding quantified by the inter spike intervals
and regular/irregular spike patterns are well captured by different reduction
schemes. In addition, for modulated stimulus with a frequency following a
sinusoidal modulation, the phase and amplitude of the PC response curve are
also well described by different reduction schemes. However, there is a
variation of coding performance across different reduction schemes for PCs
from different species. Among these reduction schemes, Branch method, which
keeps the feature of the geometrical volume of a neuron, can achieve a good
balance between different performance measures of accuracy, simplification,
computational efficiency, and spike shape change of morphology reduction, and
reproduce various phenomena shown in the full model with the whole PC
morphology. Thus, these results suggest that Branch reduction scheme could
serve as a general guideline for reducing complex morphology into a few
compartments without loss of basic characteristics of firing property of
neurons.

##  II Methods

Figure 1:  Schematic view of reduction process. (A) Illustration of a
dendritic field. Each sub-branch has a father dendrite with several child
dendrites. Each of them can be indexed as the $j$-$th$ dendrite (green) with a
set of sub-dendrites (red). (B) Illustration of different coding schemes of
neuronal morphological structure. Levels of dendrites are colored differently
with terminal dendrites as red and non-terminal dendrites as other colors. The
number of each dendrite is the order value obtained by different coding
schemes. For instance, for Branch model, the order value of non-terminal
dendrites is the sum of all their sub-dendrite order values plus a weight as
the number of sub-dendrites divided by 10. Thus, the order values of all
terminal dendrites are 1. At the next level, the dendrites have a order value
as 2.2 that is the sum of 2, from all their sub-dendrite order values, with
0.2 as a weight of the number of sub-dendrites divided by 10. So the final
order value at this level is 2.2. This process moves from the terminal or non-
terminal dendrites level by level to get all the order values. Note Elect
method operates in an opposite way. (C) Reduction process. Coding: The 1st
step is to encode morphological structure and distinguish different functional
areas (Blue, Trunk. Green, Smooth. Gray, Spiny) of a neuron. Clustering: the
2nd step is to set clusters according to the coding number. Mapping clusters:
the 3rd step is to map each cluster into a single compartment. Mapping
synapses: the 4th step is to map synaptic locations in the reduced model. The
red dots indicate synaptic locations in the spiny dendrites.

###  II-A Neuronal morphology reduction

A reduction method is to map the full morphological structure into an
equivalent reduced model with much less dendritic compartments. As a typical
tree-like structure, one has to identify the father and child dendrites, and
mark them with a set of graphic notations (see Fig. 1(A)). For the markers,
one needs to set up a coding scheme such that there is an order value for each
selected section or area as illustrated in Fig. 1(B). Here we propose four
different coding schemes where the order values are determined differently.
The motivations of these coding schemes are inspired by general analysis of
river networks [38, 39].

The main feature of Shreve and Horton methods is to identify and classify
river types based on the importance of rivers in water networks. Specifically,
Shreve coding is to define a river without tributaries as level 1, and other
river levels are obtained by adding their tributary levels. The
characteristics of the Shreve method are similar to the calculation of
confluence, and have a relatively large reference value in some simulations of
hydrological flow and sediment volume. Horton coding is to define a river
without tributaries as level 1, and other river levels are obtained by maximum
level of its tributary. Horton code mainly reflects the hierarchical
relationship in the water network and the depth of the river subtree. The
Purkinje cell dendritic structure is similar to the river network, and these
two methods can be used to classify the dendrites and then merge and simplify
the unimportant dendrites. As a result, the different functional regions of
PCs can be characterized by their order values.

Before setting the order value, the PC tree-like morphological structure can
be divided into terminal and non-terminal dendrites. We use $E$ and $O$ to
represent the set of terminal dendrites and non-terminal dendrites,
respectively. Specifically, $E=\\{e_{1},e_{2},\cdots,e_{i}\\}$ is terminal
dendrites set, $e_{i}$ is the $i$-$th$ dendrite in the set $E$, $N_{e,i}$ is
the order value of the $i$-$th$ dendrite in set $E$.
$O=\\{o_{1},o_{2},…..o_{j}\\}$ is non-terminal dendrites set, $o_{j}$ is the
$j$-$th$ dendrite in $O$, $N_{o,j}$ is the order value of the $j$-$th$
dendrite in $O$. Then according to the order value $N$, $N=N_{e,i}$ or
$N=N_{o,j}$, the dendrites can be classified as spiny, smooth, or trunk, as
following:

$dend\in\left\\{\begin{array}[]{ll}Spiny,&\textrm{if }\quad N\leq{s1}\\\
Smooth,&\textrm{if}\quad s1<N<s2\\\ Trunk,&\textrm{if}\quad
N\geq{s2}\end{array}\right.$ (1)

where $s1$ and $s2$ are the best threshold values for dividing the functional
areas of spiny, smooth, and trunk dendrites. We tested several different
values of $s1$ and $s2$, and determined the best values that can achieve the
balance between a higher reduction and a better overall accuracy. These values
of chosen for each PC are given in Table I.

In this study, we used four different coding schemes as below.

####  II-A1 Branch scheme

In this scheme, the order values of dendrites are determined by the branches
of each dendrite. The order values of terminal dendrites are fixed as 1, i.e.,
$N_{e,i}=1$, then the non-terminal dendrites order values are given as:

$N_{o,j}=\sum_{k=1}^{n}N_{o,j_{k}}+weight$ (2)

$N=\\{N_{o,j_{1}},N_{o,j_{2}},\cdots N_{o,j_{n}}\\}$ is the order value of
sub-dendrites of the $j$-$th$ dendrite in set $O$, $j_{k}$ is $k$-$th$ sub-
dendrite of the $j$-$th$ dendrite, $n$ is the number of the sub-dendrites of
the $j$-$th$ dendrite, and $weight=n/10$, where the factor 10 is the best
value for trade-off between simplification and accuracy of the modelling in
our case.

####  II-A2 Horton scheme

In this scheme, the dendritic tree is encoded by Horton analysis [38]. We set
the order value for each section. Terminal dendrites of set $E$ have the order
value as 1, $N_{e,i}=1$. The order value for non-terminal dendrites can be set
as:

$N_{o,j}=max(N_{o,j_{1}},N_{o,j_{2}},\cdots,N_{o,j_{n}})+1$ (3)

where, $N=\\{N_{o,j_{1}},N_{o,j_{2}},\cdots N_{o,j_{n}}\\}$ is the order value
of sub-dendrites of the $j$-$th$ dendrite in set $O$, $j_{n}$ is $n$-$th$ sub-
dendrite of the $j$-$th$ dendrite, and $n$ is the number of the sub-dendrites
of the $j$-$th$ dendrite.

####  II-A3 Shreve scheme

In this coding scheme, the morphological structure is quantitatively analyzed
by using the adaptive Shreve encoding scheme [39]. Terminal dendrites of set
$E$ are given by the order value as 1, $N_{e,i}=1$. The non-terminal dendrites
order values are given as:

$N_{o,j}=\sum_{k=1}^{n}N_{o,j_{k}}$ (4)

where $N_{o,j_{k}}$ is the order values of the $k$-$th$ child dendrite of the
$j$-$th$ in set $O$, and $n$ is the number of the child dendrites of the
$j$-$th$ dendrite.

####  II-A4 Elect scheme

In this scheme, different dendritic geometries result in different electricity
properties of dendrites, so one can use the feature of input resistance to
analyze tree dendrites, so that the values of the input resistance of
dendrites are used as the dendritic order values:

$N=(N_{e,i},N_{o,j})=\textrm{Input resistance}$ (5)

In this model, different functional regions of PCs can be characterized by
their order values as

$dend\in\left\\{\begin{array}[]{ll}Spiny,&\textrm{if }\quad N\geq{s2}\\\
Smooth,&\textrm{if}\quad s1<N<s2\\\ Trunk,&\textrm{if}\quad
N\leq{s1}\end{array}\right.$ (6)

Note Eq. (6) is operating in an opposite way as Eq. (1), since for Branch,
Horton and Shreve methods, tree order values are increased from spiny to
trunk, but Elect method is working as a decreasing process.

TABLE I: Best thresholds of reduction models for each cell. |  |  Guinea-pig1 |  Guinea-pig2 |  Guinea-pig3 |  Mouse1 |  Mouse2 |  Mouse3 |  Mouse4 |  Rat1 |  Rat2 |  Rat3
---|---|---|---|---|---|---|---|---|---|---|---
Branch |  s1 |  3 |  3 |  3 |  3 |  3 |  3 |  3 |  3 |  3 |  3
|  s2 |  8 |  8 |  8 |  8 |  8 |  8 |  8 |  8 |  8 |  8
Horton |  s1 |  3 |  3 |  3 |  3 |  3 |  3 |  3 |  3 |  3 |  3
|  s2 |  30 |  90 |  20 |  50 |  50 |  30 |  30 |  20 |  30 |  20
Elect |  s1 |  13 |  16 |  18 |  62 |  87 |  35 |  69 |  33 |  27 |  86
|  s2 |  17 |  17 |  19 |  63 |  88 |  36 |  70 |  34 |  29 |  87
Shreve |  s1 |  10 |  10 |  10 |  10 |  10 |  10 |  10 |  10 |  10 |  10
|  s2 |  30 |  30 |  30 |  30 |  30 |  30 |  30 |  30 |  30 |  30

Once the coding stage is finished, the rest of the work-flow process of
reducing neuronal morphology is illustrated in Fig. 1 (C). One can collect
those dendrites according to their order values into a cluster, such that
there are three sets of clusters as $C_{trunk}$ for trunk dendrites,
$C_{smooth}$ for smooth dendrites, and $C_{spiny}$ for spiny dendrites. Within
each set, there is a series of subsets of the cluster in each region of the
dendritic field. Such a subset of clusters can be mapped into one single
compartment by using the same merging rule as in Ref. [14], then ionic and
synaptic conductances are scaled by a factor to preserve membrane area [14] or
preserve volume. To preserve membrane area in Horton, Shreve, and Elect
models, we use a factor
$f_{s,j}^{eq}=\frac{\sum\nolimits_{j}s_{j}}{s_{j}^{eq}},$ where
$\sum\nolimits_{j}s_{j}$ is the sum of the membrane area of each section in
the clusters, and $s_{j}^{eq}$ is the membrane area of the equivalent
compartment. However, in Branch model, one has to preserve volume instead of
membrane area, we use a factor
$f_{v,j}^{eq}=\frac{\sum\nolimits_{j}v_{j}}{v_{j}^{eq}}$, where
$\sum\nolimits_{j}v_{j}$ is the sum of the volumes of each section in the
clusters, and $v_{j}^{eq}$ is the volume of the equivalent compartment.
Finally, synapse locations are sequentially mapped in each reduced models in
the same way as in Ref. [14].

###  II-B Multi-compartment model of Purkinje cell

To compare different reduced models with full morphology, we used multi-
compartment models based on ten detailed morphological 3D reconstructions of
Purkinje cells of guinea-pig, mouse, and rat, from public archive
neuromorpho.org. Specific capacitance was set to $0.8F/{cm}^{2}$ in the soma,
and $1.5F/{cm}^{2}$ in trunk dendrites, smooth dendrites, and spiny dendrites.
Internal axial resistivity was set to $250\Omega/cm$ similar to the values
used in Ref. [7, 40].

The same set of parameter values of passive properties, such as voltage-
dependent ionic channels, kinetic, and distribution, was used for all
morphologies of PCs. There are thirteen different types of voltage-gated ion
channels modeled, eight of which (P-type $Ca^{2+}$ channel, T-type $Ca^{2+}$
channel, class-E $Ca^{2+}$ channel, persistent $K^{+}$ channel, A-type $K^{+}$
channel, D-type $K^{+}$ channel, delayed rectifier, decay of sub-membrane
$Ca^{2+}$) were inserted into soma and dendrites. In addition, three ion
channels (fast and persistent sodium channel, anomalous rectifier channel)
were solely added to soma, and two ion channels (high-threshold calcium-
activated potassium channel, low-threshold calcium-activated potassium
channel) were solely added to dendrites [41, 42].

###  II-C Stimulation protocols

Stimulation of PC was implemented by parallel fiber (PF) inputs, where
synaptic input from each PF to PC is characterized by AMPA receptors [43].
Following the typical values estimated from experiments, a total of 1000 PF
connections for a single PC was used as previously [44]. AMPA synapses were
inserted only in the spiny dendrites with a random distribution. AMPA synapses
were modeled as a double exponential conductance change with 0.5 ms and 1.2 ms
for rising and decay time respectively [45], and the maximal synaptic
conductance was drawn from a Gaussian distribution as $\rm{5\pm{0.5}}$ nS.

In addition, PCs are affected by direct synaptic inhibition coming from
molecular layer interneurons, we used a total of 500 inhibitory connections on
a single PC. These synapses were randomly distributed on the spiny dendrites,
and modeled as $\rm GABA_{A}$ [46] with a double exponential conductance
change with 0.5 ms and 2.5 ms for rising and decay time respectively, and the
maximal synaptic conductance was drawn from a Gaussian distribution as $\rm
5\pm 0.5$ nS.

PCs aligned on the mediolateral axis receive about 175,000 PF inputs [47, 48].
This arrangement results in the hypothesis that the evoked PC firing for
temporal control of movement is encoded in the cerebellum by beams of
synchronously active PCs [49]. So in our model, PCs response were driven by
1000 synchronized PF inputs random distributed on spiny dendrites [48, 50],
i.e., simulations were run with synchronized stimulation protocol.

A single stimulation consists of a sequence of spikes containing spike times
and inter-spike intervals (ISIs), so we can generate a successive spike train
by the previous spike plus regular or irregular time intervals. There are
three types of stimulation used in this study depending on the sampling
process.

####  II-C1 Poisson process

Spike trains were modeled using a homogeneous Poisson process in which the ISI
distribution is exponential. The probability density function of ISI $\tau$ is
given by:

$p(\tau)=re^{-r\tau}$ (7)

where r is mean firing rate. The mean $<\tau>$ and standard deviation $\sigma$
of ISIs are both $1/r$.

####  II-C2 Renewal process

A simple way to generate a spike train based on renewal process is to start
with a Poisson spike train and delete all but every $k$-th spike. In this way,
a spike train is obtained with ISIs $\tau$ given by gamma probability density
function:

$p(\tau)=(k\tau)^{k}\tau^{k-1}e^{-kr\tau}/{(k-1)!}$ (8)

where $k$ is the order of gamma distribution, and $r$ is mean firing rate. The
mean $<\tau>$ and standard deviation $\sigma$ are $1/r$ and$<\tau>/\sqrt{k}$,
respectively. Here k is set to 2 through the whole study.

####  II-C3 Modulated renewal process

Recent work on the statistical modeling of neural responses has focused on
modulated renewal processes in which the spike rate is a function of the
recent spiking history. Here we modeled the modulation of the firing rate
$r(t)$ as a synaptic input frequency such that

$r(t)=Asin(2\pi{ft}+\theta)+A$ (9)

where the modulation of the firing rate is thus fully specified by its
amplitude $A$ and frequency $f$ of the sinusoidal component. Thus one can
generate a spike train that satisfies this firing rate $r(t)$ by the time-
rescaling method [51, 52, 53].

###  II-D Data analysis

The full and reduced PC morphology models were stimulated in NEURON 7.4. Data
analysis was implemented with MATLAB. Unless otherwise noted, all simulations
run 2000 ms. where data from the last 1000 ms simulation were used to analyze
the results. The time step 0.025 ms was used for all simulations. For
analysis, we used four measures to characterize the performance of the reduced
method: accuracy, simplification, efficiency and spike shape change.

####  II-D1 Accuracy

The main purpose of a reduced model is to maintain the input-output (I/O)
property on the basis of simplification of neuronal morphology. We evaluated
the accuracy of the I/O property by comparing the spike trains generated
before and after reduction [14]:

$\rm\textrm{Accuracy}=\frac{TP+TN}{TP+TN+FP+FN}$ (10)

where TP (True Positives) is the number of spikes from the full model that are
also found in the reduced model. TN (True Negatives) is the number of
intervals that the neuron does not fire in both the full and reduced models.
FP (False Positives) is the number of mismatched spikes in the reduced models.
FN (False Negatives) is the number of spikes from the full model that are not
matched in the reduced models.

####  II-D2 Simplification

Morphological simplification is the most basic requirement for a reduced
method. The full morphological structure can be seen as being made up by a
series of cylindrical segments with different lengths and diameters.
Therefore, we characterized the degree of simplification as the ratio of the
segments of morphological structure before and after reduction:

$\rm
Simplification=\frac{SEG_{\textrm{full}}-SEG_{\textrm{reduced}}}{SEG_{\textrm{full}}}$
(11)

where $\rm SEG_{\textrm{reduced}}$ is the number of the segments in reduced
morphological structure, and $\rm SEG_{\textrm{full}}$ is the number of the
segments in full morphological structure.

####  II-D3 Efficiency

Another feature of a reduced model is to improve computing efficiency, as one
wants to compute the neuronal dynamics as fast as one can, in particular in
large-scale network simulation. We evaluated this feature by the ratio between
the runtime of the full model and reduced model in the same computing
environment:

$\rm Efficiency=\frac{Runtime_{full}}{Runtime_{reduce}}$ (12)

where $\rm Runtime_{full}$ is the run time of the full model and $\rm
Runtime_{reduce}$ is the run time of the reduced model.

####  II-D4 Spike shape

In addition, we also analysis the accuracy of spike shape including spike
width and spike amplitude. We evaluated this feature by comparing the changes
in the full model and the reduce model：

$\Delta Change_{X}=|Full_{X}-Reduce_{X}|$ (13)

Where X represents spike width or spike amplitude, $Full_{X}$ and $Reduce_{X}$
are the mean spike width or spike amplitude in spike trains of the full model
and reduced model, respectively.

##  III Results

We consider four different reduction schemes to simplify the morphology of PCs
from three species, then study their effects on PC responses to synaptic
inputs. To investigate how the coding capacity of PCs is affected by
morphological structure reductions, we evaluate the performance of the reduced
model with four measures: simplification, accuracy, efficiency, and spike
shape change. Furthermore, the coding capacity of PCs, in terms of firing rate
coding, spike timing coding, and modulated firing amplitude and phase under
different input synaptic frequencies are studied as well.

###  III-A Performance evaluation of morphology reduction

Ten different PCs with different morphologies from guinea-pig, mouse, and rat
were studied. The reduced morphologies were generated by four types of
reduction schemes for each cell as shown in Fig. 2 (A). Four reducing schemes
show different simplifications for a particular morphology. However, there is
a large variety of neuronal morphology for each cell from each species.
Although the same set of parameters were used in all of ten PCs, different
morphologies result in different dynamics of hyperpolarization. Such a
difference is larger across species and smaller within the same species (Fig.
2 (B)).

Figure 2:  Detailed and simplified PC morphologies. (A) Totally ten PCs from
three species are reduced by four different schemes, Branch, Horton, Elect and
Shreve methods. Spiny dendrites in gray, smooth dendrites and initial major
branches in green. Spiny dendrites receives 1000 excitatory AMPA-type synapses
from parallel fibers (red dots). (B) Hyperpolarization phases after spiking
are different in ten full PC morphologies. Poisson stimulation at 50Hz.
Figure 3:  Spiking dynamics of full and reduced PC models. Membrane potential
traces recorded from soma of three example PCs of guinea-pig, mouse and rat
with full model (red) and four reduced models (blue). Poisson stimulation at
50Hz.

Then we focus on three example PCs, guinea-pig1 (v_e_purk1Mod), mouse1
(e4cb2a2Mod), and rat1 (p20) to illustrate the spiking dynamics of full and
reduced morphology. Throughout the study, these three example PCs are used to
represent three species respectively. In general, for all PCs, the spiking
dynamics of reduced morphology in all four schemes matches that of the full
model very well (Fig. 3).

TABLE II: Accuracy of reduced models. Values are mean $\pm$ STD calculated from 21 sets of Poisson stimulation from 10 to 1K Hz. The best reduced method for each cell is marked in bold. |  Guinea-pig1 |  Guinea-pig2 |  Guinea-pig3 |  Mouse1 |  Mouse2 |  Mouse3 |  Mouse4 |  Rat1 |  Rat2 |  Rat3
---|---|---|---|---|---|---|---|---|---|---
Branch |  $0.976\pm{0.012}$ |  $0.940\pm{0.031}$ |  $0.973\pm{0.017}$ |  $0.926\pm{0.057}$ |  $0.905\pm{0.063}$ |  $0.984\pm{0.013}$ |  $0.965\pm{0.019}$ |  $0.941\pm{0.036}$ |  $0.963\pm{0.024}$ |  $0.964\pm{0.029}$
Horton |  $0.954\pm{0.021}$ |  $0.942\pm{0.029}$ |  $0.971\pm{0.015}$ |  $0.860\pm{0.078}$ |  $0.885\pm{0.064}$ |  $0.949\pm{0.032}$ |  $0.880\pm{0.069}$ |  $0.934\pm{0.038}$ |  $0.926\pm{0.035}$ |  $0.955\pm{0.028}$
Elect |  $0.886\pm{0.021}$ |  $0.825\pm{0.048}$ |  $0.880\pm{0.040}$ |  $0.970\pm{0.019}$ |  $0.860\pm{0.063}$ |  $0.938\pm{0.029}$ |  $0.848\pm{0.047}$ |  $0.968\pm{0.015}$ |  $0.781\pm{0.074}$ |  $0.850\pm{0.052}$
Shreve |  $0.943\pm{0.023}$ |  $0.924\pm{0.035}$ |  $0.972\pm{0.012}$ |  $0.906\pm{0.054}$ |  $0.913\pm{0.052}$ |  $0.959\pm{0.020}$ |  $0.939\pm{0.033}$ |  $0.889\pm{0.064}$ |  $0.954\pm{0.017}$ |  $0.921\pm{0.050}$
Marosco |  $0.932\pm{0.029}$ |  $0.929\pm{0.029}$ |  $0.930\pm{0.027}$ |  $0.872\pm{0.092}$ |  $0.881\pm{0.059}$ |  $0.955\pm{0.022}$ |  $0.939\pm{0.033}$ |  $0.906\pm{0.051}$ |  $0.947\pm{0.019}$ |  $0.908\pm{0.057}$
TABLE III: Simplification of reduced models. |  Guinea-pig1 |  Guinea-pig2 |  Guinea-pig3 |  Mouse1 |  Mouse2 |  Mouse3 |  Mouse4 |  Rat1 |  Rat2 |  Rat3
---|---|---|---|---|---|---|---|---|---|---
Branch |  85.5% |  93.2% |  80.1% |  93.3% |  90.3% |  86.0% |  85.7% |  86.6% |  90.7% |  82.8%
Horton |  79.5% |  82.1% |  79.5% |  79.7% |  77.2% |  78.5% |  78.8% |  81.8% |  83.6% |  80.6%
Elect |  96.7% |  96.9% |  96.2% |  95.7% |  91.9% |  97.1% |  98.9% |  93.1% |  94.3% |  97.7%
Shreve |  88.8% |  90.7% |  87.6% |  91.7% |  88.8% |  89.7% |  88.5% |  93.3% |  91.7% |  91.0%
Marosco |  91.4% |  93.0% |  95.1% |  93.9% |  92.2% |  92.7% |  95.1% |  94.6% |  94.1% |  93.9%
TABLE IV: Efficiency of reduced models. Values are mean $\pm$ STD.  |  Guinea-pig1 |  Guinea-pig2 |  Guinea-pig3 |  Mouse1 |  Mouse2 |  Mouse3 |  Mouse4 |  Rat1 |  Rat2 |  Rat3
---|---|---|---|---|---|---|---|---|---|---
Branch |  $19\pm 3$ |  $22\pm 8$ |  $9\pm 2$ |  $24\pm 7$ |  $20\pm 2$ |  $15\pm 2$ |  $15\pm 3$ |  $13\pm 1$ |  $21\pm 6$ |  $12\pm 3$
Horton |  $9\pm 2$ |  $9\pm 2$ |  $8\pm 1$ |  $11\pm 1$ |  $9\pm 1$ |  $9\pm 1$ |  $10\pm 2$ |  $11\pm 2$ |  $13\pm 2$ |  $10\pm 2$
Elect |  $40\pm 8$ |  $59\pm 12$ |  $46\pm 8$ |  $56\pm 9$ |  $30\pm 4$ |  $64\pm 6$ |  $72\pm 1$ |  $16\pm 1$ |  $39\pm 13$ |  $41\pm 2$
Shreve |  $16\pm 3$ |  $18\pm 3$ |  $12\pm 1$ |  $18\pm 3$ |  $13\pm 1$ |  $14\pm 1$ |  $14\pm 2$ |  $20\pm 2$ |  $17\pm 5$ |  $16\pm 4$
Marosco |  $24\pm 6$ |  $27\pm 4$ |  $35\pm 4$ |  $31\pm 3$ |  $23\pm 2$ |  $24\pm 3$ |  $26\pm 1$ |  $22\pm 1$ |  $24\pm 4$ |  $22\pm 3$
TABLE V:  Change of spike amplitude in reduced models (mv, mean $\pm$ STD).  |  Guinea-pig1 |  Guinea-pig2 |  Guinea-pig3 |  Mouse1 |  Mouse2 |  Mouse3 |  Mouse4 |  Rat1 |  Rat2 |  Rat3
---|---|---|---|---|---|---|---|---|---|---
Branch |  $1.08\pm{0.53}$ |  $3.63\pm{1.95}$ |  $1.63\pm{0.98}$ |  $0.68\pm{0.88}$ |  $6.23\pm{2.74}$ |  $0.48\pm{0.17}$ |  $1.03\pm{0.57}$ |  $3.71\pm{1.82}$ |  $3.87\pm{1.87}$ |  $0.78\pm{0.27}$
Horton |  $2.08\pm{1.02}$ |  $3.12\pm{1.37}$ |  $1.42\pm{0.62}$ |  $1.03\pm{0.24}$ |  $7.30\pm{3.38}$ |  $0.63\pm{0.16}$ |  $0.55\pm{0.58}$ |  $1.81\pm{0.55}$ |  $0.62\pm{0.27}$ |  $0.91\pm{0.28}$
Elect |  $1.58\pm{0.80}$ |  $3.09\pm{1.36}$ |  $1.24\pm{1.00}$ |  $0.54\pm{0.22}$ |  $12.10\pm{5.89}$ |  $2.00\pm{0.74}$ |  $5.17\pm{3.30}$ |  $0.95\pm{0.50}$ |  $2.85\pm{0.70}$ |  $1.72\pm{1.43}$
Shreve |  $3.02\pm{1.40}$ |  $4.98\pm{2.60}$ |  $1.95\pm{1.02}$ |  $0.86\pm{0.19}$ |  $4.95\pm{2.49}$ |  $0.81\pm{0.26}$ |  $0.89\pm{0.65}$ |  $3.56\pm{1.32}$ |  $0.56\pm{0.18}$ |  $1.98\pm{0.53}$
Marosco |  $3.86\pm{1.88}$ |  $4.40\pm{2.18}$ |  $4.00\pm{1.83}$ |  $1.80\pm{0.48}$ |  $6.61\pm{2.89}$ |  $1.04\pm{0.30}$ |  $0.80\pm{0.61}$ |  $3.81\pm{1.34}$ |  $0.79\pm{0.24}$ |  $2.55\pm{1.24}$
TABLE VI: Change of spike width in reduced models (ms, mean $\pm$ STD).  |  Guinea-pig1 |  Guinea-pig2 |  Guinea-pig3 |  Mouse1 |  Mouse2 |  Mouse3 |  Mouse4 |  Rat1 |  Rat2 |  Rat3
---|---|---|---|---|---|---|---|---|---|---
Branch |  $0.29\pm{0.23}$ |  $0.73\pm{0.29}$ |  $0.18\pm{0.16}$ |  $0.20\pm{0.25}$ |  $1.02\pm{0.88}$ |  $0.06\pm{0.08}$ |  $0.53\pm{0.50}$ |  $0.28\pm{0.26}$ |  $0.92\pm{1.07}$ |  $0.09\pm{0.09}$
Horton |  $0.34\pm{0.27}$ |  $0.41\pm{0.32}$ |  $0.23\pm{0.15}$ |  $0.20\pm{0.18}$ |  $0.84\pm{0.44}$ |  $0.25\pm{0.27}$ |  $0.81\pm{0.32}$ |  $0.18\pm{0.19}$ |  $0.30\pm{0.19}$ |  $0.13\pm{0.18}$
Elect |  $0.81\pm{0.62}$ |  $1.39\pm{1.78}$ |  $0.44\pm{0.45}$ |  $0.26\pm{0.14}$ |  $0.51\pm{0.74}$ |  $0.34\pm{0.21}$ |  $0.4\pm{0.4}$ |  $0.11\pm{0.12}$ |  $0.30\pm{0.23}$ |  $0.49\pm{0.19}$
Shreve |  $0.42\pm{0.39}$ |  $0.59\pm{0.42}$ |  $0.27\pm{0.35}$ |  $0.23\pm{0.14}$ |  $0.92\pm{0.45}$ |  $0.21\pm{0.15}$ |  $0.55\pm{0.32}$ |  $0.37\pm{0.37}$ |  $0.28\pm{0.22}$ |  $0.36\pm{0.43}$
Marosco |  $0.49\pm{0.33}$ |  $0.52\pm{0.33}$ |  $0.41\pm{0.32}$ |  $0.31\pm{0.16}$ |  $0.57\pm{0.41}$ |  $0.18\pm{0.23}$ |  $0.31\pm{0.13}$ |  $0.34\pm{0.46}$ |  $0.32\pm{0.22}$ |  $0.47\pm{0.69}$

It is especially important to ensure the firing accuracy of the reduced model,
which can be seen in Fig. 3. The detailed measure quantified by accuracy shown
in Table II, which describes how a somatic membrane potential trace of the
full model can be reproduced in different reduced models. For comparison,
Marosco model was included [14]. The accuracy is relatively diverse in four
reduced schemes, but in general, Branch model has the highest accuracy.
However, Marosco reduced model does not archive accurate firing activities.

The second measure is simplification that characterizes how the morphologies
are changed by reduced models. Reduced model simplifies complex tree
structures into a much smaller set of dendrites that can be quantified by the
degree of simplification. Table III shows that the Elect model has the largest
degree of simplification such that the reduced morphology is most compact. In
contrast, Horton model has a minimum degree of simplification. Marosco model
has a slightly larger degree of simplification than Elect model for mouse2 and
rat1.

A direct outcome of simplification is that the simplest model needs less
computing time for simulation. It is feasible to ensure the application of
reduced models on large-scale networks only by improving computational
efficiency while high accuracy is maintained. Table IV shows the
quantification of runtime, where Elect model achieves the most efficient
computation as it has the highest degree of simplification, except that
Marosco model is fastest for rat1.

Furthermore, it is also important to maintain the accuracy of spike shape,
which can be measured by the change of spike amplitude in Table V and spike
width in Table VI, where Branch model has higher accuracy for more cells.

Given that the accuracy as the most important factor for reduction, Branch
model is the best one for the performance of accuracy of spike rate, the
degree of simplification, runtime and spike shape, are also good. The reason
for this is mainly due to the fact that the scaling factor in the
simplification process is to preserve the cell volume in Branch model, but the
other three models preserve the cell membrane area. Thus, we conclude that
cell volume may play an important role in shaping Purkinje cell firing.

###  III-B PC firing rate coding

Figure 4:  PC firing responses in full and reduced models. (A) Schematic view
of Poisson stimulation sequences from 60 Hz to 260 Hz injected to spiny
dendrites of full (left) and reduced (right) models. (B) Firing response
curves of ten PCs with full morphology. (C) and (D) Comparison of firing
response curves of three example PCs from guinea-pig, mouse and rat under four
reduction schemes, Branch, Horton, Elect and Shreve, respectively. PC curves
are grouped by species in (C) and by reduction schemes in (D). Note that there
are four PC response curves in one full model, since each is a realization of
random distribution of PF input synapses. Solid curves in (B-D) are fitted
exponential functions. Poisson stimulation in all cases.

To characterize the performance of reduced models, we carry out a wide range
of stimulation frequencies from 10Hz to 1KHz sampled in Poisson process as
illustrated in Fig. 4 (A), which is in the same range as observed in vivo
experiments where granule cells can discharge from a few Hz up to 1K Hz [54].

The cerebellum can control high-precise motor patterns with millisecond
resolution by using a wide range of action potential firing rates [7, 17, 55].
Fig. 4 (B) shows ten PCs firing rate response curves under Poisson stimuli
ranging from 10 Hz to 1K Hz in full morphology model. When such stimuli are
used, different species have their own response frequency range except for PC
rat2 that has a lower range comparing to other two rat PCs.

There is a variation of PC response curves within the same species due to
their morphological differences. The PC firing rates dramatically increase
with stimulations from 10 Hz to 180 Hz, and then slowly saturate at the higher
stimulation frequencies, which is similar to the experimental observations
found in awake animals where PCs exhibit an action potential rate between 30
and 200 Hz [56, 57].

By using the same three example PCs as in Fig. 3, one can compare the detailed
differences of PC response across different species and reduced models. Fig. 4
(B) shows that guinea-pig PC response is significantly lower than mouse and
rat, in particular, during high-frequency stimulation. This is consistent with
the difference shown in the phase of hyperpolarization after spiking (Fig.
2(B)), even other parameters were fixed at the same. This may imply that rat
and mouse are able to react on millisecond timescale activities faster than
guinea-pig.

Four different reduced methods divide the PCs into different spiny dendrites,
which leads to synaptic distribution locations slightly different, as a
result, four corresponding full models are slightly different in their
response curves (Fig. 4 (C-D)). The response frequency has to be high enough
when the stimulation frequency is high as evidenced by experimental data that
PC could generate a high stage of firing rate to control fast timing patterns
of the motor behavior. PC response curves can be well fitted by exponential
function although the high frequency is slowly saturating.

After morphology is reduced, PC response curves are preserved from the full
model, but there are some differences depending on the reduction schemes as
shown in Fig. 4 (C-D). Shreve and Horton models mediate PC firing rate
decreased compared to full model at high stimulation frequencies. This could
be seen from both mouse and rat, but guinea-pig shows slightly increased PC
firing rate. In Elect model, guinea-pig shows a significant decrease in firing
rate. In Branch model, the PC rate decreases in mouse while guinea-pig and rat
show similar firing rate. The increased diversity at high frequency
stimulation implies that complex spatial morphology structure can affect PCs
to react to millisecond-scale activity. In general, Branch model is the best
model to preserve the input-output relationship of PCs in all species.

The similar results were also observed when stimulation protocol is changed
from Poisson to renewal process(Supplementary Fig. 1). Therefore, the PC
firing responses vary depending on morphology and species rather than the
types of stimulation used.

###  III-C PC timing coding

Figure 5:  PC temporal responses under Poisson and renewal stimulation with
Branch method. (A) Schematic view of PF input (top) and PC output spike trains
from full model (middle) and Branch model (bottom) from mouse. Poisson
stimulation at 100Hz. (B) ISI distribution of spike trains from PF (gray), PC
full model (light red) and PC reduced model (green), respectively, under
Poisson and renewal process stimulation for guinea-pig (top), mouse (middle),
rat (bottom). All stimuli are at 50Hz for 10 seconds. Similarity between the
ISI distributions of full and reduced model measured by p-value, Wilcoxon
Rank-sum test. Guinea-pig, 0.51; Mouse, 0.97; Rat 0.26 for Poisson
stimulation, and Guinea-pig, 0.63; Mouse 0.93; Rat, 0.32 for renewal
stimulation. (C) Distribution of $\rm COV_{2}$ values obtained from spike
trains of PF inputs (left), PC full model (middle), and Branch model (right)
from mouse with Poisson stimulation of different frequencies. (D) $\rm
PC_{cov2}/PF_{cov2}$ showing the regularity between PF inputs and PC outputs
for full model (green) and reduced model (purple) of guinea-pig, mouse and
rat. Poisson and renewal process stimulation with different frequencies from
10 to 1K Hz.

Purkinje cells transmit precise timing information to their downstream targets
for precise control of motor-related tasks and conditioned behaviors [58, 22].
Here by using the full and reduced models with different stimulation
protocols, we investigate the effect of morphology structure on PC temporal
coding, in particular, we focus on simple spikes that are a majority of PC
spikes quantified by their inter spike intervals (ISIs), which has been shown
statistics of ISIs play an important role in PC temporal coding [59, 60].

Fig. 5 shows the results of temporal coding precision of three same PCs from
guinea-pig, mouse, and rat, under the stimulation protocols of Poisson and
renewal processes. A schematic view of PF inputs and PC outputs of the full
and reduced models illustrates that Branch model reduces the full morphology
but keeps the temporal coding very accurately (Fig. 5(A)). A common way to
characterize the temporal structure of spike train is to use the coefficient
of variation (COV) of ISIs. Based on this measure, the spike trains generated
by the PCs are significantly more regular than those of PFs. The detailed
statistics of ISIs of three example PCs with both full and reduced models
confirm this observation (Fig. 5(B)). Furthermore, the ISI distributions
obtained by the reduced model are similar to those by the full model, which
can be described by p-values from Wilcoxon Rank-sum test. For all three PCs
and both stimulation protocols, p-values of two distributions of full and
reduced model are non-significant. For the same frequency stimulus, the PC ISI
distributions are different from each other across two stimulation protocols
of Poisson and renewal process, and also different across species as well.

Fine characterization of temporal precision of spike trains can be described
by a modified measure $\rm COV_{2}=2|ISI_{n+1}-ISI_{n}|/(ISI_{n+1}+ISI_{n})$,
which can avoid that one of the ISI in the sequence affects global regularity
[61, 60]. A single spike train can obtain a sequence of $\rm COV_{2}$ values
(Supplementary Fig.3), and their distributions are shown in Fig. 5(C) for
several example frequencies of Poisson stimulation. Not surprisingly, the $\rm
COV_{2}$ values of PF stimulation sequence are distributed uniformity with
different Poisson frequencies. However, the $\rm COV_{2}$ values of PC spike
trains have a wide distribution with the peak around 0.1.

One can average all $\rm COV_{2}$ values for each spike train, then use the
value of the ratio $\rm PC_{COV_{2}}/PF_{COV_{2}}$ as an indicator to evaluate
the regularity between PF inputs and PC outputs as shown in Fig. 5(D), such
that the ratio greater than 1 indicates that PF firing is more regular,
otherwise, PC is more regular. Similar to ISI distribution, there are clear
differences across species. Changing of firing regularity is found in mouse
and rat, but not in guinea-pig. It is worth noting that PC spike trains are
more regular with the only exception that the ratio is close to 1 for renewal
process in low stimulus frequency (10 Hz) (Supplementary Fig. 2(D-F)), while
PC spike trains are always more regular for Poisson stimulation.

Figure 6:  Membrane potential traces recorded from soma of three example PCs
of guinea-pig, mouse and rat in Branch model. Poisson stimulation at 500Hz.

Furthermore, the performance of different reduced models to capture the
temporal structure of spike trains shown in the full model is different. Under
the same stimulation condition, the spike trains of mouse PC show more
regularity in Horton and Shreve models, but rat show more regularity in Branch
and Shreve models. However, guinea-pig only shows more regularity in Elect
model under renewal process stimulation. In addition, both full and reduce
models have similar results under different stimulations protocols
(Supplementary Fig. 2).

Figure 7:  PC spiking pattern statistics with Branch method. (A) Illustration
of regular and irregular spiking patterns from spike trains. Each black bar
indicates a spike. The start of regular patterns is colored in red, pink
indicates successive ISIs in regular patterns, an irregular pattern is in
green, and single ISI is in blue. Mouse PC is used. (B) Statistics of regular
pattern size across a range of Poisson and renewal process stimulation for
guinea-pig, mouse, and rat in both full and reduced models. Percentage of
different size indicated by different colors, as there are more patterns in
higher frequency. C) Percentage of regular patterns at different stimulation
frequencies. (D) Single ISI duration across a range of Poisson stimulation.
(E) Statistics of irregular pattern size across a range of Poisson
stimulation. Percentage of different size indicated by different colors.

Cerebellar PCs are observed to generate regular spike trains in vivo [61, 62].
PC simple spike trains contain highly regular spiking patterns and may
transfer information coded by regular spike patterns to downstream deep
cerebellar nuclei neurons [60].

To see this effect, we applied a wide range of stimulation frequencies to
extract the regular and irregular spiking patterns. A zoom-in illustration of
firing activity is shown in Fig. 6 under 500 Hz Poisson stimulation, which
shows that guinea-pig has most irregular refractory periods as demonstrated in
Fig. 2 where there is a much slower hyperpolarization phase for guinea-pig,
compared to mouse and rat under the same stimulus condition.

To characterize this in detail, we applied a threshold of 0.2 on the measured
$\rm COV_{2}$ values similar to the previous study [60] to extract a series of
segments of regular spiking patterns in individual spike trains obtained by
Branch model as illustrated in Fig. 7 (A).

Not surprisingly, when stimulus frequency is larger, regular patterns have
small ISIs due to increased noise (Supplementary Fig. 4). However, the
statistics of regular spiking patterns across species are quite different as
shown in Fig. 7 (C). Increasing of input frequency results in more regular
patterns for mouse and rat, but there is no significant change for guinea-pig.
Results of all four reduced models are similar to the full model for guinea-
pig. For mouse, the difference is significantly larger in Horton and Shreve
models, whereas this difference is very weak in Branch and Elect models. Rat
PC shows more regular patterns in the full model than Branch and Shreve
reduced models, but this difference is not obvious in Horton model (see
Supplementary Fig. 7).

The size of the regular pattern is defined as the number of ISIs in regular
patterns. Most of the patterns have only 2-3 ISIs for guinea-pig.
Interestingly, for mouse and rat the regular pattern sizes are widely
distributed at high frequencies, where there are quite a few longer regular
patterns in mouse as in Fig. 7 (B) and Supplementary Fig. 8.

Fig. 7 (D) shows that guinea-pig has fewer single ISI and the single ISI
duration distribution is more discrete at low frequencies. In contrary to
regular pattern, irregular pattern sizes are widely distributed at low
frequencies for mouse and rat, however, guinea-pig has larger irregular size
as in Fig. 7 (E) and Supplementary Fig. 9.

Thus, we conclude that reduced models can capture the PC temporal coding in a
reasonable range, which is in particular important for maintaining precise
timing patterns converted by PCs to theirs downstream deep cerebellar nuclei
neurons to control motor patterns.

###  III-D PC coding of modulated inputs

In the previous sections, we demonstrated input-output relationships of PC
firing activity by Poisson and renewal process stimulations with a constant
firing rate. Recently, in vivo studies show that the modulations of PC simple
spike firing are related to a number of functions in terms of behavior,
prediction, and sensory feedback [37, 63]. In order to elucidate simple spike
firing modulation, we simulated the dynamics of the PC in response to PF
inputs with modulated firing amplitude and frequency. As the activities of PFs
from granular cells are represented by modulations of stimuli, one can simply
model PF inputs as sinusoidal modulations [53] that can generate PF spike
trains from a modulated renewal process. By varying the amplitude and
frequency of sinusoidal input, we investigated the ability of firing
modulation of PC simple spikes under different reduced models.

Figure 8: PC firing modulated by sinusoidal PF inputs with different
amplitudes. (A) Illustration of PF inputs and PC outputs. PF sinusoidal input
at 1 Hz frequency with amplitude 20 (top) and 50 (bottom). Modulated PC firing
voltage traces over three cycles of input for full model and Branch model. (B)
Similar to (A) with four different amplitudes of 20, 50, 80, and 100 Hz
inputs. (top) PF input spike trains and modulated firing rate. PC spike
responses from (middle) full model and (bottom) Branch model. Mouse PC used in
(A-B). (C) Comparison of modulation amplitudes of PF input vs PC output in
full and reduced models of four reduced schemes for guinea-pig (red), mouse
(green), and rat (blue), respectively. (D) Similar to (C), but for phase
change of PC firing modulation for full model and reduced models. Sinusoidal
stimulation frequency is 1Hz in all cases.

We first set out to analysis PC firing modulation by changing PF input
amplitude, where frequency and phase of sinusoidal input is 1 Hz and 0
respectively. Fig. 8 (A) shows example results of mouse PC firing in response
to modulated PF inputs at two different amplitudes, together with responses
from reduced Branch model. The amplitude of PC modulation is expected to be
equal to that of the oscillating input when the reduced model is accurate
enough, which are shown in Fig. 8 (B) with four tightly matched example
responses for mouse PC.

For the full model, the modulated amplitudes of mouse and rat PC responses are
well matched to those of PF inputs (Fig. 8 (C)). However, guinea-pig PC is
different: well-matching is observed for lower amplitudes up to 90 Hz (80Hz in
Elect method), but smaller than PF input amplitude for larger input
amplitudes. Similarly, the PC firing of guinea-pig shows in phase with low
amplitude PF inputs, but slightly leads phase at high input amplitude (large
than 100Hz). However, mouse and rat PC modulation phases are in phase with PF
inputs for all input frequencies (Fig. 8 (D)).

To investigate the firing modulation when PC morphological structures are
changed, we simulated four corresponding reduced models. Comparing to the full
model, guinea-pig shows lower modulation amplitudes for all reduced models.
Mouse shows lower modulation amplitudes only in Branch and Horton reduced
models with higher sinusoidal amplitude (90Hz). However, the modulation
amplitudes of the rat are always matched to the PF input amplitudes in four
reduced models (Fig. 8 (C)). Moreover, in four reduced models, the modulation
phases of mouse and rat are always in phase with PF inputs. However, guinea-
pig shows leading phases at low input amplitudes and lagged phases at high
input amplitudes. Therefore when the modulation amplitude changes, there is a
large influence on PC modulated amplitudes and little influence on PC
modulated phase. In particular, for guinea-pig, the complex tree structure
plays a key role in the firing modulation.

Figure 9: PC firing modulated by sinusoidal PF inputs with different
frequencies. (A) Illustration of PF inputs and PC outputs. PF sinusoidal input
with amplitude 50 at frequencies of 0.5 Hz (top) and 1 (bottom). Modulated PC
firing voltage traces over five cycles of input for full model and Branch
model. (B) Similar to (A) with four different frequencies of 0.5, 1, 5, and 10
Hz inputs. (top) PF input spike trains and modulated firing rate. PC spike
responses from (middle) full model and (bottom) Branch model. Mouse PC used in
(A-B). (C) Modulation amplitudes of PC output in full and reduced models of
four reduced schemes for guinea-pig (red), mouse (green), and rat (blue),
respectively, at different PF input frequencies. (D) Similar to (C), but for
phase change of PC firing modulation for full model and reduced models. PF
sinusoidal stimulation amplitude is 25Hz in (B-D).

Recently, theoretical studies showed the rate of PC tonic firing could be
modulated by somatic injection of sinusoidal currents up to remarkably high
frequency (1 kHz) [7]. Here we replaced the current input with the modulated
synaptic input as above, changed the frequency of PF sinusoidal input, and
analyzed its effect on PC firing modulation as shown in Fig. 9.

Fig. 9 (A) shows example results of mouse PC firing modulation with given PF
sinusoidal inputs with a fixed amplitude and phase as 0 for both full and
Branch reduced models. PC firing is changing dynamically with different
frequency changes. Averaged PC spiking responses over many cycles of inputs
show that PC firing is well modulated and fitted with a wide range of
sinusoidal input frequencies for both full and Branch reduced models (Fig. 9
(B)) for the amplitude of 25 Hz.

PC output amplitudes can be characterized by fitted sinusoidal functions,
which shows that the PC output amplitude of this modulation is equal to the PC
input amplitude up to 3Hz for three typical PCs in the full models (Fig. 9
(C)). Then there is a small fluctuation of amplitude changes. Overall, these
results are very captured by reduced models under different schemes. The only
exception is for guinea-pig shown in Elect method (Fig. 9 (C)). In addition,
the modulation phase changes are in phase with PF inputs at most of the
frequencies (Fig. 9 (D)) for both full and reduced models across guinea-pig,
mouse, and rat.

However, when the PF input amplitude is 50Hz, the PC output amplitude can only
be modulated equally at sinusoidal low frequencies below 3Hz and decays for
larger frequencies, but the amplitude of guinea-pig decays faster than mouse
and rat (Supplementary Fig. 12). Moreover, for Elect reduced model, the PC
output amplitude is much less than the input amplitude at all frequencies
(Supplementary Fig. 12).

Therefore, the PC firing could be modulated by simplifying the morphological
structure at low amplitude when the sinusoidal frequency changes. However, the
firing rate of guinea-pig PC could not be modulated at high amplitude in Elect
reduced model when the frequency is changing. We conclude that the Elect
model, which has the highest degree of simplification, makes it difficult to
be modulated at high amplitudes. Together with the previous results, we
believe that the complex morphological structure contributes to PC firing
modulation with given sinusoidal PF inputs.

###  III-E Inhibition effect on PC coding

Purkinje cells are the sole output of the cerebellar cortex. So far, we only
consider the effects of excitatory inputs on PCs coding ability, since them
receive the only excitatory signals of parallel fibers from granule cells
which are the only output of granular layer. However, PCs also receive direct
inhibitory input from several classes of molecular layer interneurons [64, 65,
46], which contribute the firing activities of PCs significantly as well [64].

Therefore, the effect of direct inhibition on PC firing activities were also
studied by adding a population of inhibitory synaptic connections (See
Methods). Indeed, we found there is a significant change of PC firing
activities due to the inhibition input as shown in Fig. 10 with the same three
example PCs of guinea-pig, mouse and rat under four reduction scheme under
Poisson stimulation. Compared to firing activities in Fig. 4 and spike timing
patterns in Fig. 7 where there is only PF input, there is a systematic
influence from direct inhibitory input ( See Supplementary Fig. 10 and
Supplementary Fig. 11). Although the results with inhibition are comparable to
those without inhibition for most of cells, inhibitory input do play a
functional role for adjusting firing activities, similar to the experimental
observations [64]. The detailed performance with inhibitory inputs was
computed in a similar way shown in Supplemental Tab. 1 for accuracy,
Supplemental Tab. 2 for spike amplitude, and Supplemental Tab. 3 for spike
width. Again, depending on the specific cells and species, there is a large
diversity among different reduced methods.

Figure 10:  PC firing activities with 500 inhibitory and 1000 excitatory
inputs in Poisson stimulation. (A) Full (left) and Branch (right) models
receive excitatory (red) and inhibitory (blue) synapses. (B) Comparison of
firing response curves of three example PCs from guinea-pig, mouse and rate
under four reduction schemes, Branch, Horton, Elect and Shreve, respectively.
(C) Statistics of regular pattern size across a range of Poisson stimulation
for guinea-pig, mouse, and rat in full and Branch models. Percentage of
different size is indicated by different colors. (D) Similar as (C), but for
irregular patterns. (E) Single ISI duration across a range of Poisson
stimulation.

##  IV Summary and Discussion

In this study, we investigated the coding capacity of Purkinje cells to
excitatory parallel fiber input with different morphological reduction
schemes. We proposed four reduction schemes to reduce the PC complex dendritic
tree to a few components, and tested these reduced models with 10 specific
detailed PCs from three species of guinea-pig, mouse and rat respectively.

We showed by performance evaluation and simulation that reduced methods can
balance accuracy and computational efficiency in different ways. We found that
the Branch method has a better accuracy in most PCs, which is likely related
to preserved volume in the reduced method. We also found that there is no
direct relationship between accuracy and simplification. In Branch reduced
model, PCs have the lowest level of simplification but only guinea-pig2 cell
has the highest accuracy. In Elect reduced model, mouse1 cells show the
highest accuracy and simplification at the same time. Rat1 cell has the
highest degree of simplification but presents the lowest accuracy. In
addition, the computing efficiency is proportional to the degree of the
simplification. Therefore, the diversity of different reduction methods for
performance implies that one has to choose a proper method depending on the
questions to be addressed.

Compared to the Marosco model [14], we found Horton and Shreve model has a
better accuracy in most PCs, in particular, Branch model is more accuracy than
the Marosco model across all PCs. Furthermore, Elect model has a larger degree
of simplification than Marosco model in most PCs. We also found that the
accuracy of the spike shape for four reduced models is higher than Marosco
model in most PCs. However, there is no single method can achieve both good
accuracy and simplification at the same time. These suggest that there is a
trade-off of morphology reduction between accurate firing activity and
efficient runtime of stimulation, which may need more systematically
investigations at the level of neuronal network.

Precise firing coding is a key property of Purkinje cells in the cerebellum
that is mainly used to control the high-precise motor patterns with
millisecond timescale [7, 17]. We found that mouse and rat PCs can response to
generate higher firing rate than guinea-pig PCs with the same stimulation.
This may imply rats and mice are able to react on millisecond timescale better
than guinea-pig, and may have better performance for precise temporal control
of motor-related tasks and conditioned behaviors. It is worth noting that the
animal species are not uniquely represented by encoding of firing rate since
the same species have different rate coding due to their differences in
morphology. However, the detailed mechanisms of why different species show
different dynamic behaviors during reduction remain unclear. More evidence is
need to confirm and extend the current conclusion in the future.

Besides firing rate coding, timing coding can be represented in PCs spiking
patterns. With respect to simple spikes of PCs in response to PF excitatory
inputs, PCs generate more regular spike trains than the stimulation sequences
of Poisson process. For stimulation of renewal process, PF input sequences are
more regular than PC spike trains at low frequencies. In contrary, PC spike
trains are more regular than stimulation sequences at high frequencies.
Furthermore, rat and mouse PC spikes are more regular than guinea-pig at high
frequencies. It is worth noting that increasing the rate of input results in a
regularity in mouse and rat, but has no effect on guinea-pig. Inter-spike
intervals have a linear relationship with refractory periods following each of
spikes [66]. We found that guinea pig has most irregular refractory periods
under the same stimulus condition (Fig. 6). Thus, we suggest that regularity
is presumably determined by the refractory period. Moreover, mouse and rat
have similar result of the proportion of regular patterns in reduced models,
but mouse is more regular. This phenomenon is likely to be determined by the
refractory period.

We stress the importance of morphology because the morphology is known to
define the feature of neuronal types and has significant influences on
neuronal computation. A wide range of patterns of firing activity, dendritic
processing and synaptic integration produced by differences in morphology
determine the response of neurons to synaptic inputs [4, 67, 68]. Indeed, we
found differences in neuronal morphology determine the response of PCs to
synaptic inputs with a variation of firing rate, firing timing, and firing
patterns. Here we mainly consider the case of PF synaptic inputs, since adding
inhibitory inputs does not change the current results. However, it should be
noted that the dynamics of neurons in the cerebellum is driven by a large
diversity of synapses [53], and network dynamics can be reshaped by the
various types of synaptic plasticities [69, 70], future efforts are needed to
study the effect of morphology on neuronal and network dynamics via different
synaptic dynamics [7, 71, 72].

Figure 11: Complex spikes reproduced by reduced models. (A) Full and reduced
morphology of guinea-pig receiving CF (red dot) synapses. (B) Membrane
potential traces recorded from soma of guinea-pig PC in full (red) and four
reduced models (blue). The shaded part indicates complex spikes. Poisson
stimulation at 200Hz.

It is well known that Purkinje cells affect motor behavior via both simple and
complex spikes [73, 74]. We mainly addressed simple spikes in this study and
future work is needed to systematically analyze complex spikes discharged by
PC. Simple spikes occur spontaneously and are modulated by synaptic inputs
from granule cells. Complex spikes arise from climbing fibers and can induce
plasticity of other afferents to Purkinje cells, therefore it plays the
functional role of teaching or error signals during motor learning [75, 76].
The granule cell inputs modulate simple spike firing up to 200 Hz, while
climbing fibers trigger complex spikes at a remarkably low frequency (1 Hz)
[77]. In addition, complex spikes trigger prominent temporal pauses in the
simple spike trains [78]. While parallel fibers make synaptic contacts on
spines in the spiny dendrite region, climbing fiber makes synaptic contact on
the main and smooth dendrite regions [79, 80]. We mainly simplified the spiny
dendrites of the reduced models and preserved most of the morphological
structure of smooth and main dendrites. While parallel fibers make synaptic
contacts on spines in the spiny dendrite region, climbing fiber makes synaptic
contact on the main and smooth dendrite regions. However, we found membrane
potential traces can be reproduced well in four reduced models at climbing
fiber input (Fig. 11). Thus, we speculate our reduced model can reproduce the
properties of complex spikes very well. However, further systematic study is
need to investigate complex spikes.

Therefore, reduced method can be redesigned to take into account this feature
of the regional difference. It is possible that spiny dendrites have no effect
on evoked complex spikes, so they can be eliminated when simplifying the
morphological structure.

Purkinje cell is a very unique cell type in that it has a very large, perhaps
the largest and most dense neuronal morphology. However, there are many other
cell types defined in other areas of the brain, in which neuronal morphologies
are very different as well. Therefore, it is possible that one has to design
specific reduction schemes according to the uniqueness of neuronal morphology
for each cell type. For instance, the typical morphology of layer V neurons in
the cortex can generate specific dendritic spikes that are, in particular,
important to be kept when their morphologies are reduced [8]. Thus, one may
not expect that there is a unique method for morphological reduction of all
cell types in the neuronal system.

## Acknowledgments

We thank S. Jia for comments and suggestions. This work is supported by the
National Key Research and Development Program of China (Grant No.
2016YFB0800601), the National Natural Science Foundation of China (Grant Nos.
61572385 and 6161101371), the Fundamental Research Funds for the Central
Universities (Grant No. JB170309), the Science and Technology Projects of
Xi’an, China (Grant No. 201809170CX11JC12), the International Talent Exchange
Program of Beijing Municipal Commission of Science and Technology (Grant No.
Z181100001018026), and the Royal Society Newton Advanced Fellowship (Grant No.
NAF-R1-191082).

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