Chaos may enhance information transmission in theinferior olive
Nicolas Schweighofer*†, Kenji Doya*‡, Hidekazu Fukai§, Jean Vianney Chiron¶, Tetsuya Furukawa储, and Mitsuo Kawato‡*Core Research for Evolutional Science and Technology (CREST), Japan Science and Technology Corporation, and ‡ATR, Computational Neuroscience
Laboratories, 2-2-2, Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-0288, Japan; §Department of Information Science, Gifu University, Gifu 501-1193, Japan;¶Ecole Nationale Supérieure des Télécommunications, 75014 Paris, France; and 储Nara Institute of Science and Technology, Ikoma, Nara 630-0101, Japan

Despite unique well characterized neuronal properties, such as
extensive electrical coupling and low firing rates, the role of theinferior olive (IO), which is the source of the climbing fiber inputs
to cerebellar Purkinje cells, is still controversial. We propose that
the IO stochastically recodes the high-frequency information carried by its synaptic inputs into stochastic, low-rate spikes in its
climbing fiber output. Computer simulations of realistic IO networks showed that moderate electrical coupling produced chaotic
firing, which maximized the input– output mutual information.
This ‘‘chaotic resonance’’ may allow rich error signals to reachindividual Purkinje cells, even at low firing rates, allowing efficientcerebellar learning.

Over a century of cerebellar research has provided us withcomprehensive understandings of cerebellar anatomy andphysiology. It is well known, for instance, that the output neuronsof the cerebellar cortex, the Purkinje cells, receive two major
types of synaptic inputs: ⬎100,000 parallel fibers and a single
climbing fiber, an axon from an inferior olive (IO) neuron;
whereas summation of parallel fiber inputs generate ‘‘simple
spikes,’’ a single IO spike generates a ‘‘complex spike’’ throughthe powerful climbing fiber input. Furthermore, the major
anatomical and electrophysiological properties of the IO neurons are well characterized (1). First, these neurons generate
dendritic and somatic spikes at low firing rates in vivo (three
spikes per sec at most) (2). Second, they are electrotonicallycoupled by gap junctions (3), more extensively than any othercells of the mammalian brain (4). Third, they exhibit subthreshold oscillations in vitro (5, 6). However, despite this detailed
knowledge, we still lack a clear understanding of the function of
the cerebellum, and two seemingly contradictory major hypotheses have been proposed, each based on a dramatically different
view of the IO (7, 8).
According to the cerebellar learning hypothesis (9–13), when
conjointly activated with parallel fibers, IO spikes modify cerebellar input–output transformations, in agreement with the
known long-term depression (LTD) at the parallel fiber–
Purkinje cell synapse (14). Recent cerebellar motor learningtheories (12, 15–19) make two further postulates relevant to thishypothesis. First, the IO neurons must fire at a low firing rate sothat complex spikes encoding error signals do not interfere withsimple spikes carrying motor control commands (12, 15). Second, the IO must transmit error signals (15, 20) with hightemporal resolution for cerebellar learning for efficient motor
control.
Alternatively, the cerebellar timing hypothesis states that the
IO exerts its influence on motor control in real time via
synchronous and rhythmic discharges (21). This hypothesis is inagreement with the facts that IO neurons are extensively electrically coupled via gap junctions (8) and that IO neurons firewith some degree of rhythmicity and pair-wise synchrony both invitro (22) and in vivo (21). It is further supported by the generalview that electrical junctions synchronize firings of coupledneural oscillators (6, 23). Note, however, that this hypothesis
makes no mention of the widely accepted cerebellar plasticity,www.pnas.org兾cgi兾doi兾10.1073兾pnas.0305966101

such as LTD. Conversely, the cerebellar learning theories have
yet to accommodate studies showing that the IO neurons areextensively electrically coupled, and fire with some degree ofrhythmicity and pair-wise synchrony.
Here, we propose a theory of the role of the IO that is not only
coherent with all of the IO data reviewed above, but also explains
the facts that although the IO neurons themselves have low firing
rates, their inputs have high firing rates (24). We postulate thatthe major role of the IO is to reduce the firing rate carrying theerror signal for cerebellar learning while maintaining its information content. Although the observed low firing rates may
appear to contradict the assumption that the IO output carries
high-resolution error signals, recent data show that peristimulus
histograms (PSTH) of complex spikes do contain high frequency
components (Fig. 1 a and b) (15). This means that IO spikes are
generated at different timings in each trial, so that the accurate
error signal is available for individual Purkinje cells after repeated trials. Although the error signal waveform is not explicitly
reconstructed, the cumulative effects of LTD, known to havetime constants up to several hours (14), virtually average theclimbing fiber inputs (12, 15, 16) (see Discussion). LTD can thus
restore high-frequency information content of the error signal
for individual Purkinje cells. If the IO discharges were rhythmic
and synchronized, the PSTH would have a very sharp peak at theonset of a volley of synaptic inputs, and the IO would nottransmit information-rich signals (Fig. 1c). In contrast, if the IO
discharges were irregular and desynchronized, after summation,
the PSTH would be similar to the input waveform (Fig. 1d). Thatis, if the spike timing is varied on each learning episode, eachindividual Purkinje cell can restore IO input content by temporal
averaging through LTD.
How then could irregular and desynchronized firing arise incoupled IO neurons? Although strong electrical coupling synchronizes neurons, moderate electrical coupling can theoretically desynchronize neurons (25–28) and can make the discharges of individual cells irregular (25). This possibility is
consistent with the apparently random IO firing in vivo (15, 29)(see Fig. 1a for an example) and the chaotic subthreshold IO
oscillations in vitro (30). To test this hypothesis, we simulate
small networks of realistic IO cell models and study the effect ofelectrical coupling on both spiking behavior and input–output
information transmission.
Methods
Cell Model. The single IO cell model (see Fig. 2) consisted of asomatic and a dendritic compartment to represent the known
location of the ionic currents, gap junctions, and synaptic inputs(25). The somatic compartment included a low-threshold calcium current (ICal), an anomalous inward rectifier current (Ih),This paper was submitted directly (Track II) to the PNAS office.
Abbreviations: IO, inferior olive; LTD, long-term depression; PSTH, peristiumlus histograms;
OFR, ocular following response; CC, coupling coefficient.
†To whom correspondence should be addressed. E-mail: nicolas@atr.co.jp.

© 2004 by The National Academy of Sciences of the USA

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Edited by Richard F. Thompson, University of Southern California, Los Angeles, CA, and approved February 5, 2004 (received for review September 16, 2003)

Fig. 2. Two-compartment biophysical model of an IO cell. The somatic ioniccurrents include a low-threshold calcium inward current (ICal), an anomalous
inward rectifier current (Ih), the Hodgkin–Huxley type inward sodium (INa) andoutward delayed rectifier potassium (IKdr) currents, and a leakage current (Ils).
The dendritic currents include a high-threshold inward calcium current (ICah),an outward calcium-dependent potassium current (IKCa), and a leakage current (Ild). Because the dendritic potassium current IKCa is calcium-dependent,
Ca2⫹ concentration dynamics is included in the model. gint is the coupling
conductance between the two compartments. The electrotonic coupling between this cell and multiple other cells is not shown.
Fig. 1. Data showing that PSTH of complex spikes contain high-frequency
components. (a) Complex spikes recorded in a Purkinje cell during OFR in amonkey. The raster plot shows the large variability in the timing of complex
spikes from trial to trial during 40 presentations of visual stimulus motion. (b)Temporal average of 487 trials from the individual Purkinje cell shows that the
PSTH (2-msec time bins) carries high-frequency information (thin black line).
The thick red line shows that a generalized linear model can accurately
reproduce IO neuron firing rate; data are from ref. 15 with permission. (c andd) Diagrams illustrating the advantage of irregular and desynchronized firingof IO neurons. (c) A cluster of IO cells receive a common synaptic inputs of highfiring rates (Top), and they fire regularly and synchronously by electricalcoupling (Middle). Because all of the IO cells in a cluster fire synchronously at
the synaptic input onset, they do not respond to the remaining part of the
input; moreover, repetitive presentation of the same input induces very
similar responses. Thus, the PSTH of each cell, or a cluster of IO cells, exhibitsa sharp peak at the onset (Bottom). (d) On the other hand, if in response to the
input (Top), IO firing is irregular and desynchronized (Middle), the same input
induces different responses for each neuron over trials. Thus, the PSTH is moresimilar to the input waveforms (Bottom): the temporal resolution of thetransmitted information is high after temporal averaging. [b is reproducedwith permission from ref. 15 (Copyright 1998, Am. Physiol. Soc., Bethesda).]a Hodgkin–Huxley type sodium current (INa), and a delayed
rectifier potassium current (IKdr). The somatic low-thresholdcalcium current had a window of conductance around the resting
potential, which caused excitation in response to hyperpolarizing
current pulses. The dendritic compartment contained a highthreshold calcium current (ICah), a calcium-activated potassium
current (IKCa), and an electrical junction current (Ic) (25). The
dendritic high-threshold calcium current was noninactivating,which generated prolonged plateau potentials in response todepolarizing inputs. Dendritic calcium influx activated the calcium-dependent potassium current, which abruptly terminatedthe plateau potentials after 20–30 msec. Because the inactivationof this current had a long time constant, dendritic spikes were
followed by a prolonged hyperpolarization during which the cell
was less responsive to its inputs.
Our cell model was previously developed to account for in vitro
experiments. To replicate the low firing rate of IO cells andincreased excitability in vivo, as well as to incorporate new dataon electrical coupling (31), we slightly modified the model as
4656 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0305966101

follows. To decrease the firing rate, we increased the pump-out
time constant of the dendritic calcium concentrationd关Ca2⫹]⫽ ⫺1.01Cah ⫺ 0.02[Ca2⫹].
dt[1]

To increase the excitability, we used the following maximal
current conductances: gNa ⫽ 110 mS/cm2, gCal ⫽ 2 mS/cm2, gh ⫽0.15 mS/cm2; the forward rate constant of the sodium current
was changed to ␣m ⫽ (V ⫹ 48)兾1 ⫺ e(⫺(V⫹48)/3) to reduce thenumber of sodium spikes during the plateau potential. As thenonlinearity of the gap junctions seems to be more pronounced
than we previously modeled (31), we weighted the maximal
22
coupling conductance gc by f(V) ⫽ 0.8e(⫺(V /10 )) ⫹ 0.2, where V was
the membrane potentials difference between two coupled dendrites. This model adequately reproduces the coupling nonlinearity observed when one cell is strongly depolarized compared
to its neighbor (possibly because of Ca2⫹-dependent inactivationof coupling).
Synaptic Inputs. To show that irregular firing can lead to highinformation transfer of error signals to individual Purkinje cells
at low firing rates, we added, to each cell, synaptic inputs carrying
realistic error signal in an eye-movement task. Neurons in thepretectal nucleus of the optic tract innervate the IO, and exhibit
high firing rates up to 400 spikes per sec during ocular following
response (OFR) (24). Furthermore, their PSTH is similar inshape to the PSTH of complex spikes (15, 24). Thus, forsimplicity, we assumed the IO synaptic input to be the complexspike PSTH rescaled between 10 and 200 spikes per sec. OFRerror inputs, each lasting 252 msec, were presented every 1–2 sec(uniform distribution, mean 1.5 sec). This waveform was thenused as an input to 20 Poisson process spike generator: 10 werecommon to all cells and 10 were prepared separately for eachcell. These spike trains arrived at 20 excitatory synapses on eachcell.
To study the effect of noise on information transmission, eachcell received additional 20 excitatory and 20 inhibitory Poisson
Schweighofer et al.

I syn共t兲 ⫽冘gsyn共t ⫺ tk兲䡠共Vd ⫺ Esyn兲

再共t ⬍ 0兲,
共t ⬎ 0兲kg syn共t兲 ⫽0peak
gsynm䡠t䡠e⫺t/t[2]where gsynm is the maximal synaptic conductance, tk is the timeof kth spike arrival, tpeak ⫽ 2.5 msec is the time to peak, and Vd
is the dendritic membrane potential. The synaptic current reversal potential was Esyn ⫽ ⫺10 mV is for the excitatory synapsesand ⫺75 mV for the inhibitory synapses. We used this synaptic
model both for the OFR inputs (with gsynm ⫽ 0.03 mS/cm2) andfor the excitatory and inhibitory noise inputs, see below (withvarying maximal synaptic conductance).
Network Structure. To account for the variability in cellular
properties observed in real neurons, we then built small networksof heterogeneous neurons with heterogeneous coupling. In our
simulations we used two types of networks: (i) chain networks of
9 ⫻ 1 cells, in which each cell is coupled to its two nearest
neighbors (except for the two cells at the extremities); and (ii)grid networks of 2 ⫻ 2, 3 ⫻ 3, and 9 ⫻ 3 cells, where the cells
are connected to their two, three, or four neighbors dependingon their positions in the grid. To account for the variability incellular properties, the maximal conductance gCal of the low
threshold calcium current were drawn from uniform distributions, with maximum deviations set at ⫾20% of the mean (larger
deviations led to some cells not firing at all). Furthermore, thecoupling strengths gc were also taken from uniform distribution,with maximum deviation set at 20% of the mean for all networks.
A hyperpolarizing current of Ihyp ⫽ ⫺0.1 ␮A/cm2 was injected inboth the soma and the dendritic compartments.
Lyapunov Exponents. To verify whether the irregular firing waschaotic, we computed the Lyapunov exponents (32) for different
network structures and different coupling strengths. To computethe Lyapunov exponents, we simulated 10 different 2 ⫻ 2, 1 ⫻9, and 3 ⫻ 3 networks (computation of the Lyapunov exponents
was impracticable for 9 ⫻ 3 networks); each network had
different gCal and gc drawn from uniform distributions. We
started the computation of the exponents at 50 sec after an initial
perturbation and recorded the exponent values at 100 sec, whenthe values converged. We linearized the 40 or 90 dimension (10variables for each cell) system by deriving the analytical form of
Jacobian at time t, Jt, calculated the evolution of an orthonormal
set of vectors Vt and computed the exponents in terms of thelength of the principal axes.
Coupling Coefficients and Input Conductance. To examine whether
the coupling strength leading to chaos and high informationtransfer in our models were biologically realistic, we computed
the coupling coefficients (CCs), which are defined as the ratio ofchange in steady state membrane potentials of a prejunctional
cell to that of the postjunctional cell. To compute the CCs, wefirst hyperpolarized all cells to ⫺67 mV by injection of Ihyp ⫽ ⫺1
␮A/cm2. To determine the CCs, we injected a tonic depolarizing
current of 0.2 ␮A/cm2 in the soma of a ‘‘master’’ cell, andrecorded the steady state voltage change in both this master celland a neighboring cell. To best reproduce experimental conditions, we used two cells coupled to four other cells in the 3 ⫻ 9
networks. We estimated the input resistance from the voltage
response to a ⫺0.2 ␮A/cm2 current injection. The input conductance is the inverse of the input resistance.
Schweighofer et al.

Mutual Information. We quantified the influence of electricalcoupling on the low firing code of IO output by computing how
much information about the input could be extracted from theIO output spike trains, i.e., the mutual information. To measure
the mutual information robustly, we constructed 10 different
samples of 2 ⫻ 2, 3 ⫻ 3, and 3 ⫻ 9 networks; each network had
different gCal and gc drawn from uniform distributions. Then, foreach network, we ran 10 simulations (each with a different
random seed for Poisson spike generators); responses to the firsttwo OFR inputs were considered as transients and discarded. To
keep the number of spikes roughly constant, we used for analysis
the responses to 135 OFR inputs for the 2 ⫻ 2 networks, theresponses to 60 inputs for the 3 ⫻ 3 networks, and the responses
to 20 OFR inputs for the 9 ⫻ 3 networks. We detected thepresence of spikes by differentiating the membrane potential andapplying a threshold.
The mutual information was given from the formulaMI共x, y兲 ⫽ H共x兲 ⫹ H共y兲 ⫺ H共x, y兲 ⫺ C,

[3]

where x is the OFR histogram made from Poisson spikes, y is the
IO output, H(x), H(y), and H(x, y) are marginal and joint
entropies of inputs and outputs, and C is a correction termnecessary to compensate for the small number of spikes (33). We
made PSTH of both the OFR inputs and the corresponding IO
output spikes with 0.5 msec time bins, filtered them with aGaussian filter(standard deviation: 5 msec), and computed theentropies from the histograms (10 equally distant bins) of x, y,and (x, y).
Results
Properties of Single Cells and Homogenous Coupled Networks. We

previously showed that two identical coupled IO neurons could
generate desynchronized antiphase firing (25). We show herethat moderate coupling in a small 3 ⫻ 3 network of identical IO
cells, each connected to its nearest neighbors, can generate
desynchronized and irregular firing (Fig. 3). For a physiologicallyplausible coupling strength, if all of the initial conditions wereidentical, firing was synchronized and rhythmic (Fig. 3a). If the
middle cell was initially delayed only by 1 msec compared to allof the other cells, the current flowing into this cell via the gap
junctions generated a dendritic spike (first red spike in Fig. 3b),then the cells’ spikes became desynchronized. Because of theactivation of calcium-dependent potassium current the middle
cell failed to fire a somatic spike when the other cells did, andthen fired (a somatic spike) earlier than others did in the next
cycle (Fig. 3b). This shows that a coupled IO network has the twobasic mechanisms for providing chaos: (i) sensitivity to initialconditions and (ii) ‘‘expansion and folding,’’ that is, expansion ofsmall delay in spike timing and its reversal into early spiking. For
stronger coupling, a 1-msec delay in the middle cell did not
induce desynchronization but resulted in regular and periodic
firing (Fig. 3c).
Moderate Coupling Leads to Chaotic Spiking in Heterogeneous Networks. Fig. 4 shows the spiking behavior of the nine (3 ⫻ 3)

middle cells of a 9 ⫻ 3 network, which received constant current
input. Without electrical coupling (Fig. 4a), each cell generated
periodic somatic spikes. The average firing rate was 3.1 spikes persec; however, because of cellular heterogeneity, the difference infiring rates between the slowest and fastest cells was 0.7 spikes
per sec. With intermediate coupling strength (mean gc ⫽ 0.06
mS/cm2, Fig. 4b), the firing pattern of individual cell appeared
chaotic, often with one or several dendritic spikes generated
between somatic spikes, because of the long after-hyperpolarization following dendritic spikes, the average firing rate was
reduced to 1.8 spikes per sec. With strong coupling (mean gc ⫽0.5 mS/cm2; Fig. 4c), spiking became almost synchronous, and,
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synaptic inputs (adding only excitatory inputs would quickly
silent the cells by increasing the average membrane potential),
each carrying noisy inputs of mean firing rate 10 spikes per sec.
We modeled the excitatory synapses with

Fig. 3. Demonstration of the sensitivity to initial conditions for medium coupling strength for a network of identical cells. Shown are the membrane potentials
for two coupled cells in a 3 ⫻ 3 network of identical cells. Blue line, middle cell; red line, left neighbor. (a) Intermediate coupling strength (gc ⫽ 0.04 mS/cm2)and same initial conditions for all neurons: the two cells fire somatic spikes in synchrony. (b) Same coupling strength (gc ⫽ 0.04 mS/cm2), but the middle cell is
initially delayed by only 1 msec compared to all of the other cells. The middle cell (red) generates a dendritic spike as shown by the shoulder after the sodium
spike. (c) Same initial conditions as in b, but with strong coupling (gc ⫽ 0.5 mS/cm2): firing is regular (somatic spikes).

as in the uncoupled case, the cells generated only regular somatic
spikes (firing rate: 3.5 spikes per sec). This increase in synchrony,
rhythmicity, and firing rate is consistent with biological data
under conditions of increased coupling strengths (22).
To quantify the synchrony between cells, we computed themean pair-wise correlation coefficients (21) (at zero delay using
a 10-msec time bin) between the center cells and its four
immediate neighboring cells, in 3 ⫻ 3 networks. The mean
pair-wise correlation coefficients increased monotonically withcoupling: 0.012 ⫾ 0.017 for gc ⫽ 0.06 mS/cm2, 0.11 ⫾ 0.08 for gc⫽ 0.1 mS/cm2, and 0.66 ⫾ 0.053 for gc ⫽ 0.5 mS/cm2.
As seen above, moderate electrical coupling can also have theless intuitive property of making firing chaotic. For both weak (gc⬍0.06 mS/cm2) and strong (gc ⬎0.5 mS/cm2) coupling strengths,
the largest Lyapunov exponent (␭1) was close to zero for all
networks, an indication of near-rhythmic firing. For intermediate coupling strengths however the largest Lyapunov exponent
was largely positive, an indication of strong chaotic behavior(Fig. 5).
Our results offer an explanation to the seemingly contradictory reports on rhythmic (21, 22) or arhythmic (29) IO neuron
firing. It is known that the coupling strength is under control of␥-aminobutyric acid (GABA)ergic innervation that originatesfrom the cerebellar nuclei (22). When the GABAergic input ispresent in vivo (corresponding to intermediate coupling in themodel), the cells discharge with little rhythmicity and synchrony.
When the GABAergic input is blocked in vitro (corresponding tostrong coupling in the model), however, the discharges arerhythmic and synchronous. Furthermore, our results show that
although moderate electrical coupling leads to some degree ofsynchrony between cell pairs, it makes the cells arhythmic when
the whole network becomes desynchronized. This puzzling result
may be interpreted as chaotic itinerancy (34): clustering ofsubgroups of synchronized elements can vary over time incoupled systems of chaotic elements. This is consistent withmulticellular recording data showing coherence in small subgroups of cells that varied over time (35).
Moderate Coupling Increases Information Transfer. When synaptic

inputs carrying realistic error signal in an eye-movement task
were added, the overall change in firing behavior with coupling
was similar to that without inputs; without coupling, these cells
generated predominantly somatic spikes. Moderate coupling (gc⫽ 0.06 mS/cm2) yielded both somatic and dendritic spikes, and
IO discharges were irregular and desynchronized overall. Withstrong coupling, the cells only generated almost synchronizedsomatic spikes.
The mutual information per spike for a single cell at the centerof the 3 ⫻ 3 networks was 48% greater for gc ⫽ 0.06 mS/cm2 thanthat with no coupling. Furthermore, the mutual information
carried by whole networks increased by 37% in average compared to no coupling, and, because of the reduction in firing rate,
the mutual information per spike increased by 102% on average(Fig. 6). Note that the coupling strength that produced thelargest Lyapunov exponent (Fig. 5), gc ⫽ 0.1 mS/cm2, also yieldeda large information transfer. To test the generality of our results,

Fig. 4. Spiking behavior of coupled IO cells without synaptic inputs for a network of nonidentical cells. The nine cells shown are the middle 3 ⫻ 3 cells of a 9 ⫻3 network. Raster plots for respective coupling strengths gc ⫽ 0.0, 0.06, and 0.5 mS/cm2 are shown. Open circles, dendritic spikes; asterisks, somatic spikes.
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Fig. 5. Largest Lyapunov exponent as a function of coupling strength gc fordifferent networks of nonidentical cells. When the largest exponent waslarge, the network exhibited marked chaotic behavior. When the largest
exponent was close to 0, firing was almost rhythmic.

we further computed the mutual information for whole 2 ⫻ 2,
1 ⫻ 9, and 9 ⫻ 3 networks: in all cases, moderate electricalcoupling produced a large increase in mutual information per
spike (see Fig. 6).
The coupling strength leading to chaos and high informationtransfer in our models were biologically realistic, as shown by the
coupling coefficients. We found CC ⬇ 0.045 for gc ⫽ 0.06
mS/cm2. These values were compatible with experimentallyobserved CCs (between 0.002 and 0.17, with most CCs between
0.01 and 0.05) (31).
Physiological Noise Inputs Do Not Lead to Similar Increase in Information Transfer. The above results show that chaotic behavior of

coupled neurons without synaptic inputs appeared to underlie
increased input–output information transmission. This is reminiscent of ‘‘stochastic resonance,’’ a phenomenon wherein the
response of a nonlinear system to a weak input signal is optimized by the presence of a particular, nonzero level of noise (36).
For example, in threshold units such as sensory neurons, moderate noise levels improve their responses by bringing themFig. 6. Network mutual information per spike as a function of couplingstrength gc for different networks of nonidentical cells.

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Discussion
Our results suggest that electrical coupling in IO cells may inducechaos, which allow information-rich but low firing-rate error
signals to reach individual Purkinje cells, thereby providing the
cerebellar cortex with essential information for efficient learning
without disturbing on-going motor control.
The chaotic firing leads to the generation of IO spikes atdifferent timings at each trial. Because of the IO low firing, theaccurate error signal can only be available for individual Purkinjecells after repeated trials; the cumulative effects of LTD allows
the Purkinje cells to reconstruct the error signal. To illustrate
how this can be the case, let us look at the OFR system, which
has been thoroughly studied both experimentally (15) and insimulations (16, 38). An OFR movement is elicited by the sudden
motion of a large-field visual stimulus. This visual stimulus yields
a wide variety of responses in medial superior temporal (MST)
neurons (38), which are relayed to the Purkinje cells via the
granule cells. Although it is impossible to record from granulecells in vivo because of their tiny sizes, simulations showed thatthey provide Purkinje cells with a variety of temporal basis
functions (such as Gaussian radial basis functions with different
time delays), with the origin of time taken at the onset of
movement. Conjoint activity in parallel fibers and climbing fiber
induce LTD. Thus, if, for instance, one IO spike reaches a
Purkinje cell 150 msec after the stimulus motion onset, allparallel fiber synapses activated ⬇150 msec after the stimulus
motion onset will be depressed. Using the same OFR error signal
we use in the present paper, we demonstrated that the simple
spike firing temporal waveform of a Purkinje cell reconstruct,
after learning, was an approximate (but not exact, see ref. 12)
mirror image of the complex spike PSTH (16, 38).
Although we took the OFR control as an illustrative example,
our view of the IO applies to all types of movement control; in
all cases, the IO must transmit error signal with high-temporal
resolution for cerebellar learning for efficient motor control. In
particular, when the movements are rapid and need accuracy likethe saccadic or fast arm movements, a feed-forward motor
command must be accurately learned (for example, see refs. 17
and 39). The proposed chaotic firing can achieve this learning
without compromising the on-line motor command. Furthermore, our proposal also extends to the role of the cerebellum inclassical conditioning. Indeed, it has been recently proposed (40,41) that cerebellar-dependent classical conditioning, such as eye
blink conditioning (in which the animal learns how to close its
eyelid at the time of an air puff), followed the same principles as
motor control for eye movements (VOR and saccade): feedforward use of sensory information via temporally specificlearning. Furthermore, the learning rules that models LTD inboth classical conditioning and motor control are variations offunctionally similar error correcting rules; compare, for instance,
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above threshold (37). Stochastic resonance has been observed insensory systems (37), but not in the central nervous system.
Our results in 3 ⫻ 3 uncoupled networks showed that,
although the average mutual information per spike did notincrease for a maximal noise input conductance of 0.05 mS/cm2,
it increased on average 33% for 0.5 mS/cm2, and 97% for 2
mS/cm2. To examine whether 2 mS/cm2 was biologically plausible, we compared the total synaptic conductance 3,200 nS withthe cell input conductance 12.4 nS. The total synaptic conductance, consisting of 20 synapses of 2 mS/cm2, was computed fromthe surface area of the model dendritic compartment (25) as
20 ⫻ 2 mS/cm2 ⫻ 8,000 ␮m2. Given a reversal potential for
excitatory synapse Esyn ⫽ ⫺10 mV, activation of only 0.1% of thetotal synaptic conductance would produce 11.7 mV depolarization. Because this is clearly too large, we conclude that physiological ranges of synaptic conductance for noisy inputs cannot
produce large increase in mutual information.

the learning rules for classical conditioning in ref. 42 and motor
control in ref. 12. Because the classical conditioning response(the closure of the lid) is a fast and accurate motor response, theIO that carry the teaching (or error) signal should have the samerole we proposed here. Indeed, the motor command to control
the eyelid has a temporal waveform in the 200-msec range (seefigure 4.5 in ref. 40). This need for a motor command with anextended duration could, a priori, seem to run against the needfor a precise timing of the eye lid in response to the air puff;
however, simulations in ref. 40 showed that, even with an IO
input that was artificially smeared, the eye-blink occurs exactly
at the time of the air puff.
We showed here that electrical coupling could provide thesource of disorder that induced a ‘‘chaotic resonance’’ (43) in IO
networks. This resonance led to an increase in informationtransmission in IO cells by distributing the high-frequency
components of the error inputs over the sporadic, irregular andnon-phase-locked spikes by the following electrophysiologicalmechanisms. Isolated (noncoupled) IO cells tend to generate
mainly rhythmic somatic spikes for constant current inputs.
Because somatic spikes, unlike dendritic spikes, do not induce
long-lasting hyperpolarization, the firing rates are relatively high.
For very strong coupling, even inhomogeneous cells are almost
perfectly synchronized, and thus behave like a single cell.
Consequently, only somatic spikes of relatively high frequency
with rhythmicity are generated (see figure 3 in ref. 22). If phasic
synaptic inputs that contain motor error signals are fed to either

uncoupled or strongly coupled cells, the properties of the above
autonomous system (without chemical synaptic inputs) are maintained. That is, only somatic spikes are generated with relatively
high firing frequency, and those spikes tend to be phase locked
to the onset of error signals and synchronized between IO
neurons. Thus, for either no or strong coupling, informationtransmission per spike is low.
In contrast to these two cases, medium electrical coupling
induces dendritic spikes, phase jitters between neighboring
neurons, and chaos. The balance between excitatory and inhibitory currents flowing through the gap junctions was essential forchaos (see ref. 44). In the presence of even very small differencesin spike timing between coupled neurons, depolarizing currents
flowing through gap junctions can generate dendritic spikes.
Then, dendritic spikes and resulting after-hyperpolarizations
introduce complex phase shifts in neighboring neurons by excitation/inhibition, and thus induce chaos. Because chaos makes
the spikes less time-locked to error signals, mutual information
increases. It is possible to experimentally test this model bycomputing the Lyapunov exponents from recordings of spiking
IO neurons (or complex spikes) while experimentally varying theelectrical coupling strength.

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4660 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0305966101

We thank Peter Davis, Etienne Burdet, Yasushi Kobayashi, Stefan
Schaal, Frank Pollick, and Daniel Callan for comments on a previous
draft. This study was supported by the Telecommunications Advancement Organization and by Human Frontier Science Program grants(to M.K.).

Schweighofer et al.