Energy optimisation predicts the capacity of ion buffering in the brain

Neurons store energy in the ionic concentration gradients they build across their cell membrane. The amount of energy stored, and hence the work the ions can do by mixing, can be enhanced by the presence of ion buffers in extra- and intracellular space. Buffers act as sources and sinks of ions, however, and unless the buffering capacities for different ion species obey certain relationships, a complete mixing of the ions may be impeded by the physical conditions of charge neutrality and isotonicity. From these conditions, buffering capacities were calculated that enabled each ion species to mix completely. In all valid buffer distributions, the [Formula: see text] ions were buffered most, with a capacity exceeding that of [Formula: see text] and [Formula: see text] buffering by at least an order of magnitude. The similar magnitude of the (oppositely directed) [Formula: see text] and [Formula: see text] gradients made extracellular space behave as a [Formula: see text]-[Formula: see text] exchanger. Anions such as [Formula: see text] were buffered least. The great capacity of the extra- and intracellular [Formula: see text] buffers caused a large influx of [Formula: see text] ions as is typically observed during energy deprivation. These results explain many characteristics of the physiological buffer distributions but raise the question how the brain controls the capacity of its ion buffers. It is suggested that neurons and glial cells, by their great sensitivity to gradients of charge and osmolarity, respectively, sense deviations from electro-neutral and isotonic mixing, and use these signals to tune the chemical composition, and buffering capacity, of the extra- and intracellular matrices.

An Isotonic Model of Neuron Swelling Based on Co-Transport of Salt and Water

Neurons spend most of their energy building ion gradients across the cell membrane. During energy deprivation the neurons swell, and the concomitant mixing of their ions is commonly assumed to lead toward a Donnan equilibrium, at which the concentration gradients of all permeant ion species have the same Nernst potential. This Donnan equilibrium, however, is not isotonic, as the total concentration of solute will be greater inside than outside the neurons. The present theoretical paper, in contrast, proposes that neurons follow a path along which they swell quasi-isotonically by co-transporting water and ions. The final neuronal volume on the path is taken that at which the concentration of impermeant anions in the shrinking extracellular space equals that inside the swelling neurons. At this final state, which is also a Donnan equilibrium, all permeant ions can mix completely, and their Nernst potentials vanish. This final state is isotonic and electro-neutral, as are all intermediate states along this path. The path is in principle reversible, and maximizes the work of mixing.

Effect of extracellular volume on the energy stored in transmembrane concentration gradients

The amount of energy that can be retrieved from a concentration gradient across a membrane separating two compartments depends on the relative size of the compartments. Having a larger low-concentration compartment is in general beneficial. It is shown here analytically that the retrieved energy further increases when the high-concentration compartment shrinks during the mixing process, and a general formula is derived for the energy when the ratio of transported solvent to solute varies. These calculations are then applied to the interstitial compartment of the brain, which is rich in Na^{+} and Cl^{-} ions and poor in K^{+}. The reported shrinkage of this compartment, and swelling of the neurons, during oxygen deprivation is shown to enhance the energy recovered from NaCl entering the neurons. The slight loss of energy on the part of K^{+} can be compensated for by the uptake of K^{+} ions by glial cells. In conclusion, the present study proposes that the reported fluctuations in the size of the interstitial compartment of the brain (expansion during sleep and contraction during oxygen deprivation) optimize the amount of energy that neurons can store in, and retrieve from, their ionic concentration gradients.

An Interneuron Circuit Reproducing Essential Spectral Features of Field Potentials

Recent advances in engineering and signal processing have renewed the interest in invasive and surface brain recordings, yet many features of cortical field potentials remain incompletely understood. In the computational study that follows, we show that a model circuit of interneurons, coupled via both GABA_A receptor synapses and electrical synapses, reproduces many essential features of the power spectrum of local field potential (LFP) recordings, such as 1/ f power scaling at low frequency (below 10 Hz), power accumulation in the γ-frequency band (30-100 Hz), and a robust α rhythm in the absence of stimulation. The low-frequency 1/ f power scaling depends on strong reciprocal inhibition, whereas the α rhythm is generated by electrical coupling of intrinsically active neurons. As in previous studies, the γ power arises through the amplification of single-neuron spectral properties, owing to the refractory period, by parameters that favor neuronal synchrony, such as delayed inhibition. This study also confirms that both synaptic and voltage-gated membrane currents contribute substantially to the LFP and that high-frequency signals such as action potentials quickly taper off with distance. Given the ubiquity of electrically coupled interneuron circuits in the mammalian brain, they may be major determinants of the recorded potentials.

Temporal integration and 1/f power scaling in a circuit model of cerebellar interneurons

Inhibitory interneurons interconnected via electrical and chemical (GABA_A receptor) synapses form extensive circuits in several brain regions. They are thought to be involved in timing and synchronization through fast feedforward control of principal neurons. Theoretical studies have shown, however, that whereas self-inhibition does indeed reduce response duration, lateral inhibition, in contrast, may generate slow response components through a process of gradual disinhibition. Here we simulated a circuit of interneurons (stellate and basket cells) of the molecular layer of the cerebellar cortex and observed circuit time constants that could rise, depending on parameter values, to >1 s. The integration time scaled both with the strength of inhibition, vanishing completely when inhibition was blocked, and with the average connection distance, which determined the balance between lateral and self-inhibition. Electrical synapses could further enhance the integration time by limiting heterogeneity among the interneurons and by introducing a slow capacitive current. The model can explain several observations, such as the slow time course of OFF-beam inhibition, the phase lag of interneurons during vestibular rotation, or the phase lead of Purkinje cells. Interestingly, the interneuron spike trains displayed power that scaled approximately as 1/f at low frequencies. In conclusion, stellate and basket cells in cerebellar cortex, and interneuron circuits in general, may not only provide fast inhibition to principal cells but also act as temporal integrators that build a very short-term memory.NEW & NOTEWORTHY The most common function attributed to inhibitory interneurons is feedforward control of principal neurons. In many brain regions, however, the interneurons are densely interconnected via both chemical and electrical synapses but the function of this coupling is largely unknown. Based on large-scale simulations of an interneuron circuit of cerebellar cortex, we propose that this coupling enhances the integration time constant, and hence the memory trace, of the circuit.

Nonspecific synaptic plasticity improves the recognition of sparse patterns degraded by local noise

Many forms of synaptic plasticity require the local production of volatile or rapidly diffusing substances such as nitric oxide. The nonspecific plasticity these neuromodulators may induce at neighboring non-active synapses is thought to be detrimental for the specificity of memory storage. We show here that memory retrieval may benefit from this non-specific plasticity when the applied sparse binary input patterns are degraded by local noise. Simulations of a biophysically realistic model of a cerebellar Purkinje cell in a pattern recognition task show that, in the absence of noise, leakage of plasticity to adjacent synapses degrades the recognition of sparse static patterns. However, above a local noise level of 20%, the model with nonspecific plasticity outperforms the standard, specific model. The gain in performance is greatest when the spatial distribution of noise in the input matches the range of diffusion-induced plasticity. Hence non-specific plasticity may offer a benefit in noisy environments or when the pressure to generalize is strong.

Dendritic morphology predicts pattern recognition performance in multi-compartmental model neurons with and without active conductances

In this paper we examine how a neuron’s dendritic morphology can affect its pattern recognition performance. We use two different algorithms to systematically explore the space of dendritic morphologies: an algorithm that generates all possible dendritic trees with 22 terminal points, and one that creates representative samples of trees with 128 terminal points. Based on these trees, we construct multi-compartmental models. To assess the performance of the resulting neuronal models, we quantify their ability to discriminate learnt and novel input patterns. We find that the dendritic morphology does have a considerable effect on pattern recognition performance and that the neuronal performance is inversely correlated with the mean depth of the dendritic tree. The results also reveal that the asymmetry index of the dendritic tree does not correlate with the performance for the full range of tree morphologies. The performance of neurons with dendritic tapering is best predicted by the mean and variance of the electrotonic distance of their synapses to the soma. All relationships found for passive neuron models also hold, even in more accentuated form, for neurons with active membranes.

Understanding the role α7 nicotinic receptors play in dopamine efflux in nucleus accumbens

Neuronal nicotinic acetylcholine receptors (NNRs) of the α7 subtype have been shown to contribute to the release of dopamine in the nucleus accumbens. The site of action and the underlying mechanism, however, are unclear. Here we applied a circuit modeling approach, supported by electrochemical in vivo recordings, to clarify this issue. Modeling revealed two potential mechanisms for the drop in accumbal dopamine efflux evoked by the selective α7 partial agonist TC-7020. TC-7020 could desensitize α7 NNRs located predominantly on dopamine neurons or glutamatergic afferents to them or, alternatively, activate α7 NNRs located on the glutamatergic afferents to GABAergic interneurons in the ventral tegmental area. Only the model based on desensitization, however, was able to explain the neutralizing effect of coapplied PNU-120596, a positive allosteric modulator. According to our results, the most likely sites of action are the preterminal α7 NNRs controlling glutamate release from cortical afferents to the nucleus accumbens. These findings offer a rationale for the further investigation of α7 NNR agonists as therapy for diseases associated with enhanced mesolimbic dopaminergic tone, such as schizophrenia and addiction.

Impaired facial emotion recognition in patients with ventromedial prefrontal hypoperfusion

Empathy refers to our ability to recognize and share emotions by another human being. Impairment may underlie many of the emotional deficits commonly associated with a range of neuropsychiatric and neurological conditions. The prefrontal cortex (PFC) has long been implicated in these processes, but the specific contribution of subregions of the PFC remain unclear. Studies regarding the role of subregions of the prefrontal cortex such as the ventromedial prefrontal cortex (vmPFC)-in facial emotion recognition have yielded inconsistent results. The present study aimed to investigate the capacity to recognize nonverbal emotional facial expressions in a group of patients with the following: (a) perfusion deficits in the vmPFC (vmPFC group; N = 13), (b) hypoperfusions sparing the vmPFC (nonvmPFC group; N = 12), and in (c) a control group of healthy volunteers (control group; N = 17). Regions of hypoperfusion were identified by means of Single Photon Emission Computed Tomography (SPECT). Participants were asked to recognize facial expressions of the 7 basic emotions (happiness, fear, surprise, anger, disgust, sadness, or neutral). Detection of facial expressions of fear, disgust, and surprise was affected after functional disruption of the vmPFC. The present study confirms the role of the vmPFC in recognizing emotional facial expressions.

Endogenous cholinergic inputs and local circuit mechanisms govern the phasic mesolimbic dopamine response to nicotine

Nicotine exerts its reinforcing action by stimulating nicotinic acetylcholine receptors (nAChRs) and boosting dopamine (DA) output from the ventral tegmental area (VTA). Recent data have led to a debate about the principal pathway of nicotine action: direct stimulation of the DAergic cells through nAChR activation, or disinhibition mediated through desensitization of nAChRs on GABAergic interneurons. We use a computational model of the VTA circuitry and nAChR function to shed light on this issue. Our model illustrates that the α4β2-containing nAChRs either on DA or GABA cells can mediate the acute effects of nicotine. We account for in vitro as well as in vivo data, and predict the conditions necessary for either direct stimulation or disinhibition to be at the origin of DA activity increases. We propose key experiments to disentangle the contribution of both mechanisms. We show that the rate of endogenous acetylcholine input crucially determines the evoked DA response for both mechanisms. Together our results delineate the mechanisms by which the VTA mediates the acute rewarding properties of nicotine and suggest an acetylcholine dependence hypothesis for nicotine reinforcement.

Nicotine is the major addictive substance in tobacco smoke. Nicotine exerts its control over neural circuits through nicotinic acetylcholine receptors that normally respond to endogenous acetylcholine. Activation of dopamine neurons in the mesolimbic dopaminergic circuits, which signal motivational properties of actions and stimuli, is at the heart of mediating nicotine reward and dependence. However, major questions have remained unsettled over the precise mechanisms by which nicotine usurps dopaminergic signaling: through receptor activation on dopamine neurons or through receptor desensitization on local inhibitory interneurons. Here we reconcile this debate by showing that both mechanisms are possible. Most notably we present a novel hypothesis suggesting that the mechanisms for nicotine action are state-dependent; they are controlled by the rate of the endogenous cholinergic input to the dopaminergic circuits.

An integrator circuit in cerebellar cortex

The brain builds dynamic models of the body and the outside world to predict the consequences of actions and stimuli. A well-known example is the oculomotor integrator, which anticipates the position-dependent elasticity forces acting on the eye ball by mathematically integrating over time oculomotor velocity commands. Many models of neural integration have been proposed, based on feedback excitation, lateral inhibition or intrinsic neuronal nonlinearities. We report here that a computational model of the cerebellar cortex, a structure thought to implement dynamic models, reveals a hitherto unrecognized integrator circuit. In this model, comprising Purkinje cells, molecular layer interneurons and parallel fibres, Purkinje cells were able to generate responses lasting more than 10 s, to which both neuronal and network mechanisms contributed. Activation of the somatic fast sodium current by subthreshold voltage fluctuations was able to maintain pulse-evoked graded persistent activity, whereas lateral inhibition among Purkinje cells via recurrent axon collaterals further prolonged the responses to step and sine wave stimulation. The responses of Purkinje cells decayed with a time-constant whose value depended on their baseline spike rate, with integration vanishing at low (< 1 per s) and high rates (> 30 per s). The model predicts that the apparently fast circuit of the cerebellar cortex may control the timing of slow processes without having to rely on sensory feedback. Thus, the cerebellar cortex may contain an adaptive temporal integrator, with the sensitivity of integration to the baseline spike rate offering a potential mechanism of plasticity of the response time-constant.

STD-dependent and independent encoding of input irregularity as spike rate in a computational model of a cerebellar nucleus neuron

Neurons in the cerebellar nuclei (CN) receive inhibitory inputs from Purkinje cells in the cerebellar cortex and provide the major output from the cerebellum, but their computational function is not well understood. It has recently been shown that the spike activity of Purkinje cells is more regular than previously assumed and that this regularity can affect motor behaviour. We use a conductance-based model of a CN neuron to study the effect of the regularity of Purkinje cell spiking on CN neuron activity. We find that increasing the irregularity of Purkinje cell activity accelerates the CN neuron spike rate and that the mechanism of this recoding of input irregularity as output spike rate depends on the number of Purkinje cells converging onto a CN neuron. For high convergence ratios, the irregularity induced spike rate acceleration depends on short-term depression (STD) at the Purkinje cell synapses. At low convergence ratios, or for synchronised Purkinje cell input, the firing rate increase is independent of STD. The transformation of input irregularity into output spike rate occurs in response to artificial input spike trains as well as to spike trains recorded from Purkinje cells in tottering mice, which show highly irregular spiking patterns. Our results suggest that STD may contribute to the accelerated CN spike rate in tottering mice and they raise the possibility that the deficits in motor control in these mutants partly result as a pathological consequence of this natural form of plasticity.

Clustering predicts memory performance in networks of spiking and non-spiking neurons

The problem we address in this paper is that of finding effective and parsimonious patterns of connectivity in sparse associative memories. This problem must be addressed in real neuronal systems, so that results in artificial systems could throw light on real systems. We show that there are efficient patterns of connectivity and that these patterns are effective in models with either spiking or non-spiking neurons. This suggests that there may be some underlying general principles governing good connectivity in such networks. We also show that the clustering of the network, measured by Clustering Coefficient, has a strong negative linear correlation to the performance of associative memory. This result is important since a purely static measure of network connectivity appears to determine an important dynamic property of the network.

Current source density correlates of cerebellar Golgi and Purkinje cell responses to tactile input

The overall circuitry of the cerebellar cortex has been known for over a century, but the function of many synaptic connections remains poorly characterized in vivo. We used a one-dimensional multielectrode probe to estimate the current source density (CSD) of Crus IIa in response to perioral tactile stimuli in anesthetized rats and to correlate current sinks and sources to changes in the spike rate of corecorded Golgi and Purkinje cells. The punctate stimuli evoked two distinct early waves of excitation (at <10 and ∼ 20 ms) associated with current sinks in the granular layer. The second wave was putatively of corticopontine origin, and its associated sink was located higher in the granular layer than the first trigeminal sink. The distinctive patterns of granular-layer sinks correlated with the spike responses of corecorded Golgi cells. In general, Golgi cell spike responses could be linearly reconstructed from the CSD profile. A dip in simple-spike activity of coregistered Purkinje cells correlated with a current source deep in the molecular layer, probably generated by basket cell synapses, interspersed between sparse early sinks presumably generated by synapses from granule cells. The late (>30 ms) enhancement of simple-spike activity in Purkinje cells was characterized by the absence of simultaneous sinks in the granular layer and by the suppression of corecorded Golgi cell activity, pointing at inhibition of Golgi cells by Purkinje axon collaterals as a likely mechanism of late Purkinje cell excitation.

The first second: models of short-term memory traces in the brain

Many network models in computational neuroscience rise to the challenge of explaining behavioural phenomena ranging from microseconds to tens of seconds using components operating mostly on a time-scale of milliseconds. These models have in common that the underlying system has a memory, which implies that its output depends on its past input history. In this review we compare how such memory traces or delayed responses may be implemented in different brain areas supporting a diversity of functions.

Mechanism of spontaneous and self-sustained oscillations in networks connected through axo-axonal gap junctions

Electrical coupling of clusters of neurons via axo-axonal gap junctions is a candidate mechanism underlying the ultra-fast (> 100 Hz) oscillations recorded in various in vitro and in vivo normal and pathological conditions [Traub et al. (1999)Neuroscience, 92, 407-426]. The poor characterization of axo-axonal gap junctions, however, limits experimental verification of this mechanism. We simulated networks of prototype multi-compartmental neurons in order to identify the parameters constraining the production and frequency of ultra-fast oscillations. Weak axo-axonal coupling was found to synchronize networks preferentially at the gamma-range frequency (30-100 Hz). Networks with strong axo-axonal coupling were able to produce 200 Hz oscillations, a finding we extended with several new observations. Ultra-fast oscillations arose spontaneously during dendritic excitation, i.e. in the absence of extrinsic axonal background spikes, as the spike trigger zone concomitantly shifted from soma to axon. The all-or-none oscillations could be transitory or self-sustained, and lasted longer in larger networks. They occurred more likely during low to modest soma firing rates, as strong afterhyperpolarizing currents tended to impair them. The rate of the rhythm was independent of network size and of the level of excitation, but inversely proportional to the distance of the junctions from the soma. As a matter of fact, the resulting axonal firing rate was the highest one at which antidromic spikes would not collide with spikes reflected from the soma. Taken together, the observed model dynamics lends further credibility to axo-axonal coupling as a mechanism of ultra-fast oscillations.

Dendritic amplification of inhibitory postsynaptic potentials in a model Purkinje cell

In neurons with large dendritic arbors, the postsynaptic potentials interact in a complex manner with active and passive membrane properties, causing not easily predictable transformations during the propagation from synapse to soma. Previous theoretical and experimental studies in both cerebellar Purkinje cells and neocortical pyramidal neurons have shown that voltage-dependent ion channels change the amplitude and time-course of postsynaptic potentials. We investigated the mechanisms involved in the propagation of inhibitory postsynaptic potentials (IPSPs) along active dendrites in a model of the Purkinje cell. The amplitude and time-course of IPSPs recorded at the soma were dependent on the synaptic distance from the soma, as predicted by passive cable theory. We show that the effect of distance on the amplitude and width of the IPSP was significantly reduced by the dendritic ion channels, whereas the rise time was not affected. Somatic IPSPs evoked by the activation of the most distal synapses were up to six times amplified owing to the presence of voltage-gated channels and the IPSP width became independent of the covered distance. A transient deactivation of the Ca(2+) channels and the Ca(2+)-dependent K(+) channels, triggered by the hyperpolarization following activation of the inhibitory synapse, was found to be responsible for these dynamics. Nevertheless, the position of activated synapses had a marked effect on the Purkinje cell firing pattern, making stellate cells and basket cells most suitable for controlling the firing rate and spike timing, respectively, of their target Purkinje cells.

The Effect of NMDA Receptors on Gain Modulation

The ability of individual neurons to modulate the gain of their input-output function is important for information processing in the brain. In a recent study (Mitchell & Silver, 2003), shunting inhibition was found to modulate the gain of cerebellar granule cells subjected to simulated currents through AMPA receptor synapses. Here we investigate the effect on gain modulation resulting from adding the currents mediated by NMDA receptors to a compartmental model of the granule cell. With only AMPA receptors, the changes in gain induced by shunting inhibition decreased gradually with the average firing rate of the afferent mossy fibers. With NMDA receptors present, this decrease was more rapid, therefore narrowing the bandwidth of mossy fiber firing rates available for gain modulation. The deterioration of gain modulation was accompanied by a reduced variability of the input current and saturation of NMDA receptors. However, when the output of the granule cell was plotted as a function of the average input current instead of the input firing frequency, both models showed very similar response curves and comparable gain modulation. We conclude that NMDA receptors do not directly impair gain control by shunting inhibition, but the effective bandwidth decreases as a consequence of the increased total charge transfer.

Synaptic pathways in neural microcircuits

The functions performed by different neural microcircuits depend on the anatomical and physiological properties of the various synaptic pathways connecting neurons. Neural microcircuits across various species and brain regions are similar in terms of their repertoire of neurotransmitters, their synaptic kinetics, their short-term and long-term plasticity, and the target-specificity of their synaptic connections. However, microcircuits can be fundamentally different in terms of the precise recurrent design used to achieve a specific functionality. In this review, which is part of the TINS Microcircuits Special Feature, we compare the connectivity designs in spinal, hippocampal, neocortical and cerebellar microcircuits, and discuss the different computational challenges that each microcircuit faces.

Deletion of FMR1 in Purkinje Cells Enhances Parallel Fiber LTD, Enlarges Spines, and Attenuates Cerebellar Eyelid Conditioning in Fragile X Syndrome

Absence of functional FMRP causes Fragile X syndrome. Abnormalities in synaptic processes in the cerebral cortex and hippocampus contribute to cognitive deficits in Fragile X patients. So far, the potential roles of cerebellar deficits have not been investigated. Here, we demonstrate that both global and Purkinje cell-specific knockouts of Fmr1 show deficits in classical delay eye-blink conditioning in that the percentage of conditioned responses as well as their peak amplitude and peak velocity are reduced. Purkinje cells of these mice show elongated spines and enhanced LTD induction at the parallel fiber synapses that innervate these spines. Moreover, Fragile X patients display the same cerebellar deficits in eye-blink conditioning as the mutant mice. These data indicate that a lack of FMRP leads to cerebellar deficits at both the cellular and behavioral levels and raise the possibility that cerebellar dysfunctions can contribute to motor learning deficits in Fragile X patients.

Oscillations in the cerebellar cortex: a prediction of their frequency bands

Local recurrent connections endow the cerebellar cortex with an intrinsic dynamics. We performed computer simulations to predict the frequency bands of the oscillations that will most likely emerge. Feedback inhibition from the Golgi to the granule cells induced 10-50 Hz oscillations, the period at resonance being approximately equal to four times the maximum conduction delay generated along the parallel-fiber connections from granule to Golgi cells. In the molecular layer, the interneurons tended to induce fast oscillations (100-250 Hz), having a period equal to about four times the delay over their reciprocal synaptic connections. Finally, although the presence of lateral inhibition among the Purkinje cells has not been firmly established, reciprocal Purkinje-cell synapses are predicted to transform the cerebellar cortex into a potential temporal integrator.

Inactivation of calcium-binding protein genes induces 160 Hz oscillations in the cerebellar cortex of alert mice

Oscillations in neuronal populations may either be imposed by intrinsically oscillating pacemakers neurons or emerge from specific attributes of a distributed network of connected neurons. Calretinin and calbindin are two calcium-binding proteins involved in the shaping of intraneuronal Ca2+ fluxes. However, although their physiological function has been studied extensively at the level of a single neuron, little is known about their role at the network level. Here we found that null mutations of genes encoding calretinin or calbindin induce 160 Hz local field potential oscillations in the cerebellar cortex of alert mice. These oscillations reached maximum amplitude just beneath the Purkinje cell bodies and are reinforced in the cerebellum of mice deficient in both calretinin and calbindin. Purkinje cells fired simple spikes phase locked to the oscillations and synchronized along the parallel fiber axis. The oscillations reversibly disappeared when gap junctions or either GABA(A) or NMDA receptors were blocked. Cutaneous stimulation of the whisker region transiently suppressed the oscillations. However, the intrinsic somatic excitability of Purkinje cells recorded in slice preparation was not significantly altered in mutant mice. Functionally, these results suggest that 160 Hz oscillation emerges from a network mechanism combining synchronization of Purkinje cell assemblies through parallel fiber excitation and the network of coupled interneurons of the molecular layer. These findings demonstrate that subtle genetically induced modifications of Ca2+ homeostasis in specific neuron types can alter the observed dynamics of the global network.

Resonant synchronization in heterogeneous networks of inhibitory neurons

Brain rhythms arise through the synchronization of neurons and their entrainment in a regular firing pattern. In this process, networks of reciprocally connected inhibitory neurons are often involved, but what mechanism determines the oscillation frequency is not completely understood. Analytical studies predict that the emerging frequency band is primarily constrained by the decay rate of the unitary IPSC. We observed a new phenomenon of resonant synchronization in computer-simulated networks of inhibitory neurons in which the synaptic current has a delayed onset, reflecting finite spike propagation and synaptic transmission times. At the resonant level of network excitation, all neurons fire synchronously and rhythmically with a period approximately four times the mean delay of the onset of the inhibitory synaptic current. The amplitude and decay time constant of the synaptic current have relatively minor effects on the emerging frequency band. By varying the axonal delay of the inhibitory connections, networks with a realistic synaptic kinetics can be tuned to frequencies from 40 to >200 Hz. This resonance phenomenon arises in heterogeneous networks with, on average, as few as five connections per neuron. We conclude that the delay of the synaptic current is the primary parameter controlling the oscillation frequency of inhibitory networks and propose that delay-induced synchronization is a mechanism for fast brain rhythms that depend on intact inhibitory synaptic transmission.

Peripheral stimuli excite coronal beams of Golgi cells in rat cerebellar cortex

Cerebellar granule cells constitute the largest neurone population of the brain. Their axons run as parallel fibres along the coronal axis, and the one-dimensional spread of excitation that is expected to result from this arrangement is a key assumption of theories of cerebellar function. In many studies using various techniques, however, it was not possible to evoke such a beam-like propagation of excitation with natural stimuli. We recorded, in Crus I and II of anaesthetised rats, pairs of Golgi cells aligned along the parallel fibre axis and synchronising spontaneously. Each pair was subjected to two stimulation protocols: punctate and semi-continuous. Local punctate facial stimulation evoked distinct fast and late responses of variable strength and latency (fast: 4.0-10.2 ms; late: 13.6-22.7 ms). Semi-continuous stimulation with a brush increased the firing rate, and modified the precision and phase of synchronisation. Differences between a pair in response strength and phase to brush stimulation correlated strongly with the difference in latency to punctate stimulation. These observations were reproduced in a model of the granular layer. The stimulus activated a central patch of mossy fibres, and Golgi cells received short- and long-range excitation from mossy and parallel fibres, respectively. The strength and latency of the punctate response of a model Golgi cell were found to vary with its position, reflecting a systematic change in the contribution of mossy and parallel fibres to its excitation with distance from the activated patch. During brush stimulation, model Golgi cells inside the patch fired more precisely synchronised, whereas the other Golgi cells responded with a lag proportional to their distance from the patch, thereby reproducing the experimentally observed changes in synchronisation. Taken together with the previously reported large receptive fields of Golgi cells and with their spontaneous synchronisation, the variable, position-dependent latency of evoked Golgi cell responses indicates a beam-like spread of excitation along the parallel fibres in rat cerebellar cortex.

Weak common parallel fibre synapses explain the loose synchrony observed between rat cerebellar golgi cells

1. In anaesthetized rats, pairs of cerebellar Golgi cells fired synchronously at rest, provided they were aligned along the parallel fibre axis. The observed synchrony was much less precise, however, than that which would be expected to result from common, monosynaptic parallel fibre excitation. 2. To explain this discrepancy, the precision and frequency of spike synchronization (i.e. the width and area of the central peak on the spike train cross-correlogram) were computed in a generic model for varying input, synaptic and neuronal parameters. 3. Correlation peaks between model neurons became broader, and peak area smaller, when the number of afferents increased and each single synapse decreased proportionally in strength. Peak width was inversely proportional to firing rate, but independent of the percentage of shared afferents. Peak area, in contrast, scaled with the percentage of shared afferents but was almost firing rate independent. 4. Broad correlation peaks between pairs of model neurons resulted from the loose spike timing between single model neurons and their afferents. This loose timing reflected a need for long-term synaptic integration to fire the neurons. Model neurons could accomplish this through firing rate adaptation mediated by a Ca2+-activated K+ channel. 5. We conclude that loose synchrony may be entirely explained by shared input from monosynaptic, non-synchronized afferents. The inverse relationship between peak width and firing rate allowed us to distinguish common parallel fibre input from firing rate covariance as a primary cause of loose synchrony between cerebellar Golgi cells in anaesthetized rats.

Parallel fibers synchronize spontaneous activity in cerebellar Golgi cells

Cerebellar Golgi cells inhibit their afferent interneurons, the excitatory granule cells. Such a feedback inhibition causes both inhibitory and excitatory neurons in the circuit to synchronize. Our modeling work predicts that the long granule cell axons, the parallel fibers, entrain many Golgi cells and their afferent granule cells in a single synchronous rhythm. Spontaneous activity of 42 pairs of putative Golgi cells was recorded in anesthetized rats to test these predictions. In 25 of 26 pairs of Golgi cells that were positioned along the transverse axis, and presumed to receive common parallel fiber input, spontaneous activity showed a high level of coherence (mean Z score > 6). Conversely, 12 of 16 Golgi cell pairs positioned along the parasagittal axis (no common parallel fiber input) were not synchronized; 4 of 16 of them showed only low levels of synchronicity (mean Z score < 4). For transverse pairs the accuracy of the coherence, measured as the width at half-height of the central peak of the cross-correlogram, was rather low (29.8 +/- 12.5 msec) but increased with Golgi cell firing rate, as predicted by the model. These results suggest that in addition to their role as gain controllers, cerebellar Golgi cells may control the timing of granule cell spiking.

Synchronization of golgi and granule cell firing in a detailed network model of the cerebellar granule cell layer

The granular layer of the cerebellum has a disproportionately large number of excitatory (granule cells) versus inhibitory neurons (Golgi cells). Its synaptic organization is also unique with a dense reciprocal innervation between granule and Golgi cells but without synaptic contacts among the neurons of either population. Physiological recordings of granule or Golgi cell activity are scarce, and our current thinking about the way the granular layer functions is based almost exclusively on theoretical considerations. We computed the steady-state activity of a large-scale model of the granular layer of the rat cerebellum. Within a few tens of milliseconds after the start of random mossy fiber input, the populations of Golgi and granule cells became entrained in a single synchronous oscillation, the basic frequency of which ranged from 10 to 40 Hz depending on the average rate of firing in the mossy fiber population. The long parallel fibers ensured, through alpha-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid-mediated synapses, a coherent excitation of Golgi cells, while the regular firing of each Golgi cell synchronized all granule cells within its axonal radius through transient activation of their gamma-aminobutyric acid-A (GABAA) receptor synapses. Individual granule cells often remained silent during a few successive oscillation cycles so that their average firing rates, which could be quite variable, reflected the average activities of their mossy fiber afferents. The synchronous, rhythmic firing pattern was robust over a broad range of biologically realistic parameter values and to parameter randomization. Three conditions, however, made the oscillations more transient and could desynchronize the entire network in the end: a very low mossy fiber activity, a very dominant excitation of Golgi cells through mossy fiber synapses (rather than through parallel fiber synapses), and a tonic activation of granule cell GABAA receptors (with an almost complete absence of synaptically induced inhibitory postsynaptic currents). These three conditions were associated with a reduction in the parallel fiber activity, and synchrony could be restored by increasing the mossy fiber firing rate. The model predicts that, under conditions of strong mossy fiber input to the cerebellum, Golgi cells do not only control the strength of parallel fiber activity but also the timing of the individual spikes. Provided that their parallel fiber synapses constitute an important source of excitation, Golgi cells fire rhythmically and synchronized with granule cells over large distances along the parallel fiber axis. According to the model, the granular layer of the cerebellum is desynchronized when the mossy fiber firing rate is low.

The cerebellum: cortical processing and theory

Several advances over the past year have made necessary a complete re-evaluation of the function of the cerebellum and of the role of cerebellar synaptic plasticity. These advances include the discovery of parallel fiber induced long-term depression, the presence of normal motor coordination in the absence of cerebellar long-term depression in knock-out mice, and the strong activation of the cerebellar nuclei while sensory tasks are performed.

Model circuit of spiking neurons generating directional selectivity in simple cells

1. We here consider the property of directional selectivity (DS) in simple cells of layer 4 of cat area 17 as an instance of a receptive field (RF) transformation between two monosynaptically connected neuron populations: the afferent geniculate (lateral geniculate nucleus, LGN) cells and their target, layer 4 simple cells. We have studied this particular RF transformation because the large set of experimental data available allowed us to restrain the synaptic organization of our model layer 4 circuitry. 2. The one-compartment, spiking model neurons of the layer 4 circuitry are excitatory (adapting) or inhibitory (nonadapting). They all have simple-cell RFs composed of two spatially separated ON and OFF subregions. The sequence of the subregions across the neurons' RFs, which is determined by the geniculocortical inputs they receive, varies independently from their preferred direction of stimulus motion, which is determined by spatial asymmetries in their corticocortical inputs. 3. Synaptic transmission in the model layer 4 circuitry is mediated via non-N-methyl-D-aspartate (non-NMDA) receptors (geniculocortical excitation), via NMDA receptors (corticocortical excitation), and via gamma-aminobutyric acid-A receptors (corticocortical inhibition). Excitatory and inhibitory cortical neurons receive the same afferents. However, excitatory neurons form efferent synapses exclusively with neurons having the same RF characteristics, and preferentially with those having the same RF position. Inhibitory neurons form synapses preferentially with neurons having different RF characteristics or adjacent RF positions. 4. By comparing the neurons' numerically computed responses to visual stimuli with those of actual simple cells, the topology of the corticocortical connections has been constrained. The experimental responses to stationary and moving, and to bar as well as grating, stimuli are consistently reproduced with a single constant parameter setting. 5. Subsequently, the model has been analyzed from a system-theoretic approach and has been manipulated in order to find the components critical for its proper functioning. Variations on the model have been simulated for evaluating the performance of alternative connection schemes. 6. Spatially opponent inhibition between model simple cells with antagonistic RF subregions is necessary for the restoration of linearity lost at the LGN output. It hyperpolarizes model simple cells when the contrast polarity of an efficient stimulus is reversed and prevents, particularly in directionally nonselective cells, a frequency doubling of the responses to sine wave gratings of low spatial frequencies. 7. Directionally opponent inhibition between model simple cells preferring opposite directions of motion is necessary for the generation of genuine DS (a ratio of firing rates > 2 for opposite directions of motion). 8. The corticocortical excitatory polysynaptic feedback loops in the model are able to provide the time delays needed to generate DS, and even to preserve DS at very low speeds. The strength spatial extension, and time course of this corticocortical feedback excitation, together with the dynamics of the geniculate afferents and the width of the RF, determine the tuning of model simple cells in the temporal and velocity domain. 9. The present model generates directionally selective responses to stimulus motion over distances smaller than the width of a single subregion and as small as the spacing between the afferent geniculate RFs. The direction-selective mechanism acts uniformly across the entire width of a subregion. Thus the position invariance of DS arises in the present model at the same level as DS itself. The same holds for the stimulus polarity (light vs. dark) invariance of DS. Consequently, there is no need for highly hierarchical models in which all these characteristics accumulate in simple cells by pooling from lower-order subunits or neurons.