Recent data on the cerebellum require new models and theories.

The cerebellum has been a popular topic for theoretical studies because its structure was thought to be simple. Since David Marr and James Albus related its function to motor skill learning and proposed the Marr-Albus cerebellar learning model, this theory has guided and inspired cerebellar research. In this review, we summarize the theoretical progress that has been made within this framework of error-based supervised learning. We discuss the experimental progress that demonstrates more complicated molecular and cellular mechanisms in the cerebellum as well as new cell types and recurrent connections. We also cover its involvement in diverse non-motor functions and evidence of other forms of learning. Finally, we highlight the need to explain these new experimental findings into an integrated cerebellar model that can unify its diverse computational functions.

Sodium channel slow inactivation normalizes firing in axons with uneven conductance distributions.

The Na+ channels that are important for action potentials show rapid inactivation, a state in which they do not conduct, although the membrane potential remains depolarized.1,2 Rapid inactivation is a determinant of millisecond-scale phenomena, such as spike shape and refractory period. Na+ channels also inactivate orders of magnitude more slowly, and this slow inactivation has impacts on excitability over much longer timescales than those of a single spike or a single inter-spike interval.3,4,5,6,7,8,9,10 Here, we focus on the contribution of slow inactivation to the resilience of axonal excitability11,12 when ion channels are unevenly distributed along the axon. We study models in which the voltage-gated Na+ and K+ channels are unevenly distributed along axons with different variances, capturing the heterogeneity that biological axons display.13,14 In the absence of slow inactivation, many conductance distributions result in spontaneous tonic activity. Faithful axonal propagation is achieved with the introduction of Na+ channel slow inactivation. This "normalization" effect depends on relations between the kinetics of slow inactivation and the firing frequency. Consequently, neurons with characteristically different firing frequencies will need to implement different sets of channel properties to achieve resilience. The results of this study demonstrate the importance of the intrinsic biophysical properties of ion channels in normalizing axonal function.

Neuronal morphology enhances robustness to perturbations of channel densities.

Biological neurons show significant cell-to-cell variability but have the striking ability to maintain their key firing properties in the face of unpredictable perturbations and stochastic noise. Using a population of multi-compartment models consisting of soma, neurites, and axon for the lateral pyloric neuron in the crab stomatogastric ganglion, we explore how rebound bursting is preserved when the 14 channel conductances in each model are all randomly varied. The coupling between the axon and other compartments is critical for the ability of the axon to spike during bursts and consequently determines the set of successful solutions. When the coupling deviates from a biologically realistic range, the neuronal tolerance of conductance variations is lessened. Thus, the gross morphological features of these neurons enhance their robustness to perturbations of channel densities and expand the space of individual variability that can maintain a desired output pattern.

Interactions among diameter, myelination, and the Na/K pump affect axonal resilience to high-frequency spiking.

Axons reliably conduct action potentials between neurons and/or other targets. Axons have widely variable diameters and can be myelinated or unmyelinated. Although the effect of these factors on propagation speed is well studied, how they constrain axonal resilience to high-frequency spiking is incompletely understood. Maximal firing frequencies range from ∼1 Hz to >300 Hz across neurons, but the process by which Na/K pumps counteract Na+ influx is slow, and the extent to which slow Na+ removal is compatible with high-frequency spiking is unclear. Modeling the process of Na+ removal shows that large-diameter axons are more resilient to high-frequency spikes than are small-diameter axons, because of their slow Na+ accumulation. In myelinated axons, the myelinated compartments between nodes of Ranvier act as a "reservoir" to slow Na+ accumulation and increase the reliability of axonal propagation. We now find that slowing the activation of K+ current can increase the Na+ influx rate, and the effect of minimizing the overlap between Na+ and K+ currents on spike propagation resilience depends on complex interactions among diameter, myelination, and the Na/K pump density. Our results suggest that, in neurons with different channel gating kinetic parameters, different strategies may be required to improve the reliability of axonal propagation.

The Cellular Electrophysiological Properties Underlying Multiplexed Coding in Purkinje Cells.

Neuronal firing patterns are crucial to underpin circuit level behaviors. In cerebellar Purkinje cells (PCs), both spike rates and pauses are used for behavioral coding, but the cellular mechanisms causing code transitions remain unknown. We use a well-validated PC model to explore the coding strategy that individual PCs use to process parallel fiber (PF) inputs. We find increasing input intensity shifts PCs from linear rate-coders to burst-pause timing-coders by triggering localized dendritic spikes. We validate dendritic spike properties with experimental data, elucidate spiking mechanisms, and predict spiking thresholds with and without inhibition. Both linear and burst-pause computations use individual branches as computational units, which challenges the traditional view of PCs as linear point neurons. Dendritic spike thresholds can be regulated by voltage state, compartmentalized channel modulation, between-branch interaction and synaptic inhibition to expand the dynamic range of linear computation or burst-pause computation. In addition, co-activated PF inputs between branches can modify somatic maximum spike rates and pause durations to make them carry analog signals. Our results provide new insights into the strategies used by individual neurons to expand their capacity of information processing.SIGNIFICANCE STATEMENT Understanding how neurons process information is a fundamental question in neuroscience. Purkinje cells (PCs) were traditionally regarded as linear point neurons. We used computational modeling to unveil their electrophysiological properties underlying the multiplexed coding strategy that is observed during behaviors. We demonstrate that increasing input intensity triggers localized dendritic spikes, shifting PCs from linear rate-coders to burst-pause timing-coders. Both coding strategies work at the level of individual dendritic branches. Our work suggests that PCs have the ability to implement branch-specific multiplexed coding at the cellular level, thereby increasing the capacity of cerebellar coding and learning.

Firing rate-dependent phase responses of Purkinje cells support transient oscillations.

Both spike rate and timing can transmit information in the brain. Phase response curves (PRCs) quantify how a neuron transforms input to output by spike timing. PRCs exhibit strong firing-rate adaptation, but its mechanism and relevance for network output are poorly understood. Using our Purkinje cell (PC) model, we demonstrate that the rate adaptation is caused by rate-dependent subthreshold membrane potentials efficiently regulating the activation of Na+ channels. Then, we use a realistic PC network model to examine how rate-dependent responses synchronize spikes in the scenario of reciprocal inhibition-caused high-frequency oscillations. The changes in PRC cause oscillations and spike correlations only at high firing rates. The causal role of the PRC is confirmed using a simpler coupled oscillator network model. This mechanism enables transient oscillations between fast-spiking neurons that thereby form PC assemblies. Our work demonstrates that rate adaptation of PRCs can spatio-temporally organize the PC input to cerebellar nuclei.

Exploring Impaired SERCA Pump-Caused Alternation Occurrence in Ischemia.

Impaired sarcoplasmic reticulum (SR) calcium transport ATPase (SERCA) gives rise to Ca2+ alternans and changes of the Ca2+release amount. These changes in Ca2+ release amount can reveal the mechanism underlying how the interaction between Ca2+ release and Ca2+ uptake induces Ca2+ alternans. This study of alternans by calculating the values of Ca2+ release properties with impaired SERCA has not been explored before. Here, we induced Ca2+ alternans by using an impaired SERCA pump under ischemic conditions. The results showed that the recruitment and refractoriness of the Ca2+ release increased as Ca2+ alternans occurred. This indicates triggering Ca waves. As the propagation of Ca waves is linked to the occurrence of Ca2+ alternans, the "threshold" for Ca waves reflects the key factor in Ca2+ alternans development, and it is still controversial nowadays. We proposed the ratio between the diastolic network SR (NSR) Ca content (Cansr) and the cytoplasmic Ca content (Ca i ) (Cansr/Ca i ) as the "threshold" of Ca waves and Ca2+ alternans. Diastolic Cansr, Ca i , and their ratio were recorded at the onset of Ca2+ alternans. Compared with certain Cansr and Ca i , the "threshold" of the ratio can better explain the comprehensive effects of the Ca2+ release and the Ca2+ uptake on Ca2+ alternans onset. In addition, these ratios are related with the function of SERCA pumps, which vary with different ischemic conditions. Thus, values of these ratios could be used to differentiate Ca2+ alternans from different ischemic cases. This agrees with some experimental results. Therefore, the certain value of diastolic Cansr/Ca i can be the better "threshold" for Ca waves and Ca2+ alternans.

Climbing Fibers Provide Graded Error Signals in Cerebellar Learning.

The cerebellum plays a critical role in coordinating and learning complex movements. Although its importance has been well recognized, the mechanisms of learning remain hotly debated. According to the classical cerebellar learning theory, depression of parallel fiber synapses instructed by error signals from climbing fibers, drives cerebellar learning. The uniqueness of long-term depression (LTD) in cerebellar learning has been challenged by evidence showing multi-site synaptic plasticity. In Purkinje cells, long-term potentiation (LTP) of parallel fiber synapses is now well established and it can be achieved with or without climbing fiber signals, making the role of climbing fiber input more puzzling. The central question is how individual Purkinje cells extract global errors based on climbing fiber input. Previous data seemed to demonstrate that climbing fibers are inefficient instructors, because they were thought to carry "binary" error signals to individual Purkinje cells, which significantly constrains the efficiency of cerebellar learning in several regards. In recent years, new evidence has challenged the traditional view of "binary" climbing fiber responses, suggesting that climbing fibers can provide graded information to efficiently instruct individual Purkinje cells to learn. Here we review recent experimental and theoretical progress regarding modulated climbing fiber responses in Purkinje cells. Analog error signals are generated by the interaction of varying climbing fibers inputs with simultaneous other synaptic input and with firing states of targeted Purkinje cells. Accordingly, the calcium signals which trigger synaptic plasticity can be graded in both amplitude and spatial range to affect the learning rate and even learning direction. We briefly discuss how these new findings complement the learning theory and help to further our understanding of how the cerebellum works.

Voltage- and Branch-Specific Climbing Fiber Responses in Purkinje Cells.

Climbing fibers (CFs) provide instructive signals driving cerebellar learning, but mechanisms causing the variable CF responses in Purkinje cells (PCs) are not fully understood. Using a new experimentally validated PC model, we unveil the ionic mechanisms underlying CF-evoked distinct spike waveforms on different parts of the PC. We demonstrate that voltage can gate both the amplitude and the spatial range of CF-evoked Ca2+ influx by the availability of K+ currents. This makes the energy consumed during a complex spike (CS) also voltage dependent. PC dendrites exhibit inhomogeneous excitability with individual branches as computational units for CF input. The variability of somatic CSs can be explained by voltage state, CF activation phase, and instantaneous CF firing rate. Concurrent clustered synaptic inputs affect CSs by modulating dendritic responses in a spatially precise way. The voltage- and branch-specific CF responses can increase dendritic computational capacity and enable PCs to actively integrate CF signals.

Role of CaMKII and PKA in Early Afterdepolarization of Human Ventricular Myocardium Cell: A Computational Model Study.

Early afterdepolarization (EAD) plays an important role in arrhythmogenesis. Many experimental studies have reported that Ca2+/calmodulin-dependent protein kinase II (CaMKII) and ß-adrenergic signaling pathway are two important regulators. In this study, we developed a modified computational model of human ventricular myocyte to investigate the combined role of CaMKII and ß-adrenergic signaling pathway on the occurrence of EADs. Our simulation results showed that (1) CaMKII overexpression facilitates EADs through the prolongation of late sodium current's (INaL) deactivation progress; (2) the combined effect of CaMKII overexpression and activation of ß-adrenergic signaling pathway further increases the risk of EADs, where EADs could occur at shorter cycle length (2000 ms versus 4000 ms) and lower rapid delayed rectifier K+ current (IKr) blockage (77% versus 85%). In summary, this study computationally demonstrated the combined role of CaMKII and ß-adrenergic signaling pathway on the occurrence of EADs, which could be useful for searching for therapy strategies to treat EADs related arrhythmogenesis.

Cellular mechanism of cardiac alternans: an unresolved chicken or egg problem.

T-wave alternans, a specific form of cardiac alternans, has been associated with the increased susceptibility to cardiac arrhythmias and sudden cardiac death (SCD). Plenty of evidence has related cardiac alternans at the tissue level to the instability of voltage kinetics or Ca(2+) handling dynamics at the cellular level. However, to date, none of the existing experiments could identify the exact cellular mechanism of cardiac alternans due to the bi-directional coupling between voltage kinetics and Ca(2+) handling dynamics. Either of these systems could be the origin of alternans and the other follows as a secondary change, therefore making the cellular mechanism of alternans a difficult chicken or egg problem. In this context, theoretical analysis combined with experimental techniques provides a possibility to explore this problem. In this review, we will summarize the experimental and theoretical advances in understanding the cellular mechanism of alternans. We focus on the roles of action potential duration (APD) restitution and Ca(2+) handling dynamics in the genesis of alternans and show how the theoretical analysis combined with experimental techniques has provided us a new insight into the cellular mechanism of alternans. We also discuss the possible reasons of increased propensity for alternans in heart failure (HF) and the new possible therapeutic targets. Finally, according to the level of electrophysiological recording techniques and theoretical strategies, we list some critical experimental or theoretical challenges which may help to determine the origin of alternans and to find more effective therapeutic targets in the future.

Fibroblast proliferation alters cardiac excitation conduction and contraction: a computational study.

In this study, the effects of cardiac fibroblast proliferation on cardiac electric excitation conduction and mechanical contraction were investigated using a proposed integrated myocardial-fibroblastic electromechanical model. At the cellular level, models of the human ventricular myocyte and fibroblast were modified to incorporate a model of cardiac mechanical contraction and cooperativity mechanisms. Cellular electromechanical coupling was realized with a calcium buffer. At the tissue level, electrical excitation conduction was coupled to an elastic mechanics model in which the finite difference method (FDM) was used to solve electrical excitation equations, and the finite element method (FEM) was used to solve mechanics equations. The electromechanical properties of the proposed integrated model were investigated in one or two dimensions under normal and ischemic pathological conditions. Fibroblast proliferation slowed wave propagation, induced a conduction block, decreased strains in the fibroblast proliferous tissue, and increased dispersions in depolarization, repolarization, and action potential duration (APD). It also distorted the wave-front, leading to the initiation and maintenance of re-entry, and resulted in a sustained contraction in the proliferous areas. This study demonstrated the important role that fibroblast proliferation plays in modulating cardiac electromechanical behaviour and which should be considered in planning future heart-modeling studies.

Theoretical investigation of the mechanism of heart failure using a canine ventricular cell model: especially the role of up-regulated CaMKII and SR Ca(2+) leak.

Heart failure (HF) is associated with susceptibility to sudden cardiac death. However, the underlying mechanism of electrical instability and mechanical dysfunction associated with HF remains poorly understood. In this study, a new canine ventricular cell model based on the Hund-Rudy dynamic (HRd) model and recently published experimental data was developed to investigate the electrical changes and calcium handling dysfunction in HF. Simulation results suggest that: 1) acute Ca(2+)/calmodulin-dependent protein kinase II (CaMKII) over-expression (CaMKII-OE) affects the action potential (AP) profile, while AP prolongation is mainly caused by the down-regulation of K(+) currents; 2) enhanced late Na(+) current (I(NaL)) alone cannot adequately lead to [Na(+)] elevation in HF; 3) enhanced sarcoplasmic reticulum (SR) leak current (I(leak)) causes disturbed Ca(2+) handling and there is little contribution from Na(+)/Ca(2+) exchanger (NCX); 4) at high SR Ca(2+) load, a steeper fractional SR Ca(2+) release is observed in HF than that in control, causing alternans to occur more easily; and 5) I(leak) block restores the contraction and relaxation function, but cannot eliminate alternans. By inhibiting CaMKII, alternans is eliminated, but contractility is not improved. Partial CaMKII inhibition in combination with I(leak) block could augment mechanical function and depress alternans, suggesting a new possible therapeutic target for HF treatment.

A study of mechanical optimization strategy for cardiac resynchronization therapy based on an electromechanical model.

An optimal electrode position and interventricular (VV) delay in cardiac resynchronization therapy (CRT) improves its success. However, the precise quantification of cardiac dyssynchrony and magnitude of resynchronization achieved by biventricular (BiV) pacing therapy with mechanical optimization strategies based on computational models remain scant. The maximum circumferential uniformity ratio estimate (CURE) was used here as mechanical optimization index, which was automatically computed for 6 different electrode positions based on a three-dimensional electromechanical canine model of heart failure (HF) caused by complete left bundle branch block (CLBBB). VV delay timing was adjusted accordingly. The heart excitation propagation was simulated with a monodomain model. The quantification of mechanical intra- and interventricular asynchrony was then investigated with eight-node isoparametric element method. The results showed that (i) the optimal pacing location from maximal CURE of 0.8516 was found at the left ventricle (LV) lateral wall near the equator site with a VV delay of 60 ms, in accordance with current clinical studies, (ii) compared with electrical optimization strategy of E(RMS), the LV synchronous contraction and the hemodynamics improved more with mechanical optimization strategy. Therefore, measures of mechanical dyssynchrony improve the sensitivity and specificity of predicting responders more. The model was subject to validation in future clinical studies.