Determinants of synaptic integration and heterogeneity in rebound firing explored with data-driven models of deep cerebellar nucleus cells

Significant inroads have been made to understand cerebellar cortical processing but neural coding at the output stage of the cerebellum in the deep cerebellar nuclei (DCN) remains poorly understood. The DCN are unlikely to just present a relay nucleus because Purkinje cell inhibition has to be turned into an excitatory output signal, and DCN neurons exhibit complex intrinsic properties. In particular, DCN neurons exhibit a range of rebound spiking properties following hyperpolarizing current injection, raising the question how this could contribute to signal processing in behaving animals. Computer modeling presents an ideal tool to investigate how intrinsic voltage-gated conductances in DCN neurons could generate the heterogeneous firing behavior observed, and what input conditions could result in rebound responses. To enable such an investigation we built a compartmental DCN neuron model with a full dendritic morphology and appropriate active conductances. We generated a good match of our simulations with DCN current clamp data we recorded in acute slices, including the heterogeneity in the rebound responses. We then examined how inhibitory and excitatory synaptic input interacted with these intrinsic conductances to control DCN firing. We found that the output spiking of the model reflected the ongoing balance of excitatory and inhibitory input rates and that changing the level of inhibition performed an additive operation. Rebound firing following strong Purkinje cell input bursts was also possible, but only if the chloride reversal potential was more negative than −70 mV to allow de-inactivation of rebound currents. Fast rebound bursts due to T-type calcium current and slow rebounds due to persistent sodium current could be differentially regulated by synaptic input, and the pattern of these rebounds was further influenced by HCN current. Our findings suggest that active properties of DCN neurons could play a crucial role for signal processing in the cerebellum.     

Glutamate-Bound NMDARs Arising from In Vivo-like Network Activity Extend Spatio-temporal Integration in a L5 Cortical Pyramidal Cell Model

In vivo, cortical pyramidal cells are bombarded by asynchronous synaptic input arising from ongoing network activity. However, little is known about how such ‘background’ synaptic input interacts with nonlinear dendritic mechanisms. We have modified an existing model of a layer 5 (L5) pyramidal cell to explore how dendritic integration in the apical dendritic tuft could be altered by the levels of network activity observed in vivo. Here we show that asynchronous background excitatory input increases neuronal gain and extends both temporal and spatial integration of stimulus-evoked synaptic input onto the dendritic tuft. Addition of fast and slow inhibitory synaptic conductances, with properties similar to those from dendritic targeting interneurons, that provided a ‘balanced’ background configuration, partially counteracted these effects, suggesting that inhibition can tune spatio-temporal integration in the tuft. Excitatory background input lowered the threshold for NMDA receptor-mediated dendritic spikes, extended their duration and increased the probability of additional regenerative events occurring in neighbouring branches. These effects were also observed in a passive model where all the non-synaptic voltage-gated conductances were removed. Our results show that glutamate-bound NMDA receptors arising from ongoing network activity can provide a powerful spatially distributed nonlinear dendritic conductance. This may enable L5 pyramidal cells to change their integrative properties as a function of local network activity, potentially allowing both clustered and spatially distributed synaptic inputs to be integrated over extended timescales.     

Equivalence transformations for dendritic Y-junctions: a new definition of dendritic sub-unit

A sequence of equivalence transformations is used to represent the mathematical model of a simply branched neuron with non-homogeneous membrane properties and non-uniform geometry by an entirely equivalent model of an unbranched structure. The analysis indicates how neuronal morphology, in combination with its biophysical properties, shapes neuronal output in response to current input. The equivalence transformations described here reveal the types of operations that are likely to feature in the analysis of complex multi-branched structures, neuronal or otherwise. These transformations provide a new definition of dendritic sub-unit and a basis of a mechanism for characterising local and non-local signal processing within dendritic structures. It is anticipated that the capacity to transform biological neurons into an equivalent unbranched structure will make an important contribution to the understanding of the functional role of neuron geometry as well as to the construction of silicon neurons with realistic biological properties.     

Dendritic I h Selectively Blocks Temporal Summation of Unsynchronized Distal Inputs in CA1 Pyramidal Neurons

The active dendritic conductances shape the input-output properties of many principal neurons in different brain regions, and the various ways in which they regulate neuronal excitability need to be investigated to better understand their functional consequences. Using a realistic model of a hippocampal CA1 pyramidal neuron, we show a major role for the hyperpolarization-activated current, Ih, in regulating the spike probability of a neuron when independent synaptic inputs are activated with different degrees of synchronization and at different distances from the soma. The results allowed us to make the experimentally testable prediction that the Ih in these neurons is needed to reduce neuronal excitability selectively for distal unsynchronized, but not for synchronized, inputs.     

Correlated Conductance Parameters in Leech Heart Motor Neurons Contribute to Motor Pattern Formation

Neurons can have widely differing intrinsic membrane properties, in particular the density of specific conductances, but how these contribute to characteristic neuronal activity or pattern formation is not well understood. To explore the relationship between conductances, and in particular how they influence the activity of motor neurons in the well characterized leech heartbeat system, we developed a new multi-compartmental Hodgkin-Huxley style leech heart motor neuron model. To do so, we evolved a population of model instances, which differed in the density of specific conductances, capable of achieving specific output activity targets given an associated input pattern. We then examined the sensitivity of measures of output activity to conductances and how the model instances responded to hyperpolarizing current injections. We found that the strengths of many conductances, including those with differing dynamics, had strong partial correlations and that these relationships appeared to be linked by their influence on heart motor neuron activity. Conductances that had positive correlations opposed one another and had the opposite effects on activity metrics when perturbed whereas conductances that had negative correlations could compensate for one another and had similar effects on activity metrics.     

Branch Input Resistance and Steady Attenuation for Input to One Branch of a Dendritic Neuron Model

Mathematical solutions and numerical illustrations are presented for the steady-state distribution of membrane potential in an extensively branched neuron model, when steady electric current is injected into only one dendritic branch. Explicit expressions are obtained for input resistance at the branch input site and for voltage attenuation from the input site to the soma; expressions for AC steady-state input impedance and attenuation are also presented. The theoretical model assumes passive membrane properties and the equivalent cylinder constraint on branch diameters. Numerical examples illustrate how branch input resistance and steady attenuation depend upon the following: the number of dendritic trees, the orders of dendritic branching, the electrotonic length of the dendritic trees, the location of the dendritic input site, and the input resistance at the soma. The application to cat spinal motoneurons, and to other neuron types, is discussed. The effect of a large dendritic input resistance upon the amount of local membrane depolarization at the synaptic site, and upon the amount of depolarization reaching the soma, is illustrated and discussed; simple proportionality with input resistance does not hold, in general. Also, branch input resistance is shown to exceed the input resistance at the soma by an amount that is always less than the sum of core resistances along the path from the input site to the soma.     

Effects of active versus passive dendritic membranes on the transfer properties of a simulated neuron

In a simulated neuron with a dendritic tree, the relative effects of active and passive dendritic membranes on transfer properties were studied. The simulations were performed by means of a digital computer. The computations calculated the changes in transmembrane voltages of many compartments over time as a function of other biophysical variables. These variables were synaptic input intensity, critical firing threshold, rate of leakage of current across the membrane, and rate of longitudinal current spread between compartments. For both passive and active dendrites, the transfer properties of the soma studied for different rates of longitudinal current spread. With low rates of current spread, graded changes in firing threshold produced correspondingly graded changes in output discharge. With high rates of current spread, the neuron became a bistable operator where spiking was enhanced if the threshold was below a certain level and suppressed if the threshold was above that level. Since alterations in firing threshold were shown to have the same effect on firing rate as alterations in synaptic input intensity, the neuron can be said to change from graded to contrast-enhancing in its response to stimuli of different intensities. The presence or absence of dendritic spiking was found to have a significant effect on the integrative properties of the simulated neuron. In particular, contrast enhancement was considerably more pronounced in neurons with passive than with active dendrites in that somatic spike rates reached a higher maximum when dendrites were passive. With active dendrites, a less intense input was needed to initiate somatic spiking than with passive dendrites because a distal dendritic spike could easily propagate by means of longitudinal current spread to the soma. Once somatic spiking was initiated, though, spike rates tended to be lower with active than with passive dendrites because the soma recovered more slowly from its post-spike refractory period if it was also influenced by refractory periods in the dendrites. The experiment of comparing neurons with active and passive dendrites was repeated at a different, higher value of synaptic input. The same differences in transfer properties between the active and passive cases emerged as before. Spiking patterns in neurons with active dendrites were also affected by the time distribution of synaptic inputs. In a previous study, inputs had been random over both space and time, varying about a predetermined mean, whereas in the present study, inputs were random over space but uniform over time. When inputs were made uniform over time, spiking became more difficult to initiate and the transition from graded to bistable response became less sharp.     

Active Dendrites Enhance Neuronal Dynamic Range

Since the first experimental evidences of active conductances in dendrites, most neurons have been shown to exhibit dendritic excitability through the expression of a variety of voltage-gated ion channels. However, despite experimental and theoretical efforts undertaken in the past decades, the role of this excitability for some kind of dendritic computation has remained elusive. Here we show that, owing to very general properties of excitable media, the average output of a model of an active dendritic tree is a highly non-linear function of its afferent rate, attaining extremely large dynamic ranges (above 50 dB). Moreover, the model yields double-sigmoid response functions as experimentally observed in retinal ganglion cells. We claim that enhancement of dynamic range is the primary functional role of active dendritic conductances. We predict that neurons with larger dendritic trees should have larger dynamic range and that blocking of active conductances should lead to a decrease in dynamic range.     

Stochastic Ion Channel Gating in Dendritic Neurons: Morphology Dependence and Probabilistic Synaptic Activation of Dendritic Spikes

Neuronal activity is mediated through changes in the probability of stochastic transitions between open and closed states of ion channels. While differences in morphology define neuronal cell types and may underlie neurological disorders, very little is known about influences of stochastic ion channel gating in neurons with complex morphology. We introduce and validate new computational tools that enable efficient generation and simulation of models containing stochastic ion channels distributed across dendritic and axonal membranes. Comparison of five morphologically distinct neuronal cell types reveals that when all simulated neurons contain identical densities of stochastic ion channels, the amplitude of stochastic membrane potential fluctuations differs between cell types and depends on sub-cellular location. For typical neurons, the amplitude of membrane potential fluctuations depends on channel kinetics as well as open probability. Using a detailed model of a hippocampal CA1 pyramidal neuron, we show that when intrinsic ion channels gate stochastically, the probability of initiation of dendritic or somatic spikes by dendritic synaptic input varies continuously between zero and one, whereas when ion channels gate deterministically, the probability is either zero or one. At physiological firing rates, stochastic gating of dendritic ion channels almost completely accounts for probabilistic somatic and dendritic spikes generated by the fully stochastic model. These results suggest that the consequences of stochastic ion channel gating differ globally between neuronal cell-types and locally between neuronal compartments. Whereas dendritic neurons are often assumed to behave deterministically, our simulations suggest that a direct consequence of stochastic gating of intrinsic ion channels is that spike output may instead be a probabilistic function of patterns of synaptic input to dendrites.     

A model for the polarization of neurons by extrinsically applied electric fields.

A model is presented for the subthreshold polarization of a neuron by an applied electric field. It gives insight into how morphological features of a neuron affect its polarizability. The neuronal model consists of one or more extensively branched dendritic trees, a lumped somatic impedance, and a myelinated axon with nodes of Ranvier. The dendritic trees branch according to the 3/2-power rule of Rall, so that each tree has an equivalent cylinder representation. Equations for the membrane potential at the soma and at the nodes of Ranvier, given an arbitrary specified external potential, are derived. The solutions determine the contributions made by the dendritic tree and the axon to the net polarization at the soma. In the case of a spatially constant electric field, both the magnitude and sign of the polarization depend on simple combinations of parameters describing the neuron. One important combination is given by the ratio of internal resistances for longitudinal current spread along the dendritic tree trunk and along the axon. A second is given by the ratio between the DC space constant for the dendritic tree trunk and the distance between nodes of Ranvier in the axon. A third is given by the product of the electric field and the space constant for the trunk of the dendritic tree. When a neuron with a straight axon is subjected to a constant field, the membrane potential decays exponentially with distance from the soma. Thus, the soma seems to be a likely site for action potential initiation when the field is strong enough to elicit suprathreshold polarization. In a simple example, the way in which orientation of the various parts of the neuron affects its polarization is examined. When an axon with a bend is subjected to a spatially constant field, polarization is focused at the bend, and this is another likely site for action potential initiation.     

Phenomenological Incorporation of Nonlinear Dendritic Integration Using Integrate-and-Fire Neuronal Frameworks

It has been discovered recently in experiments that the dendritic integration of excitatory glutamatergic inputs and inhibitory GABAergic inputs in hippocampus CA1 pyramidal neurons obeys a simple arithmetic rule as , where , and are the respective voltage values of the summed somatic potential, the excitatory postsynaptic potential (EPSP) and the inhibitory postsynaptic potential measured at the time when the EPSP reaches its peak value. Moreover, the shunting coefficient in this rule only depends on the spatial location but not the amplitude of the excitatory or inhibitory input on the dendrite. In this work, we address the theoretical issue of how much the above dendritic integration rule can be accounted for using subthreshold membrane potential dynamics in the soma as characterized by the conductance-based integrate-and-fire (I&F) model. Then, we propose a simple I&F neuron model that incorporates the spatial dependence of the shunting coefficient by a phenomenological parametrization. Our analytical and numerical results show that this dendritic-integration-rule-based I&F (DIF) model is able to capture many experimental observations and it also yields predictions that can be used to verify the validity of the DIF model experimentally. In addition, the DIF model incorporates the dendritic integration effects dynamically and is applicable to more general situations than those in experiments in which excitatory and inhibitory inputs occur simultaneously in time. Finally, we generalize the DIF neuronal model to incorporate multiple inputs and obtain a similar dendritic integration rule that is consistent with the results obtained by using a realistic neuronal model with multiple compartments. This generalized DIF model can potentially be used to study network dynamics that may involve effects arising from dendritic integrations.     

Efficient fitting of conductance-based model neurons from somatic current clamp

Estimating biologically realistic model neurons from electrophysiological data is a key issue in neuroscience that is central to understanding neuronal function and network behavior. However, directly fitting detailed Hodgkin?CHuxley type model neurons to somatic membrane potential data is a notoriously difficult optimization problem that can require hours/days of supercomputing time. Here we extend an efficient technique that indirectly matches neuronal currents derived from somatic membrane potential data to two-compartment model neurons with passive dendrites. In consequence, this approach can fit semi-realistic detailed model neurons in a few minutes. For validation, fits are obtained to model-derived data for various thalamo-cortical neuron types, including fast/regular spiking and bursting neurons. A key aspect of the validation is sensitivity testing to perturbations arising in experimental data, including sampling rates, inadequately estimated membrane dynamics/channel kinetics and intrinsic noise. We find that maximal conductance estimates and the resulting membrane potential fits diverge smoothly and monotonically from near-perfect matches when unperturbed. Curiously, some perturbations have little effect on the error because they are compensated by the fitted maximal conductances. Therefore, the extended current-based technique applies well under moderately inaccurate model assumptions, as required for application to experimental data. Furthermore, the accompanying perturbation analysis gives insights into neuronal homeostasis, whereby tuning intrinsic neuronal properties can compensate changes from development or neurodegeneration.     

The Role of Ongoing Dendritic Oscillations in Single-Neuron Dynamics

The dendritic tree contributes significantly to the elementary computations a neuron performs while converting its synaptic inputs into action potential output. Traditionally, these computations have been characterized as both temporally and spatially localized. Under this localist account, neurons compute near-instantaneous mappings from their current input to their current output, brought about by somatic summation of dendritic contributions that are generated in functionally segregated compartments. However, recent evidence about the presence of oscillations in dendrites suggests a qualitatively different mode of operation: the instantaneous phase of such oscillations can depend on a long history of inputs, and under appropriate conditions, even dendritic oscillators that are remote may interact through synchronization. Here, we develop a mathematical framework to analyze the interactions of local dendritic oscillations and the way these interactions influence single cell computations. Combining weakly coupled oscillator methods with cable theoretic arguments, we derive phase-locking states for multiple oscillating dendritic compartments. We characterize how the phase-locking properties depend on key parameters of the oscillating dendrite: the electrotonic properties of the (active) dendritic segment, and the intrinsic properties of the dendritic oscillators. As a direct consequence, we show how input to the dendrites can modulate phase-locking behavior and hence global dendritic coherence. In turn, dendritic coherence is able to gate the integration and propagation of synaptic signals to the soma, ultimately leading to an effective control of somatic spike generation. Our results suggest that dendritic oscillations enable the dendritic tree to operate on more global temporal and spatial scales than previously thought; notably that local dendritic activity may be a mechanism for generating on-going whole-cell voltage oscillations.     

Thin dendrites of cerebellar interneurons confer sublinear synaptic integration and a gradient of short-term plasticity

Interneurons are critical for neuronal circuit function, but how their dendritic morphologies and membrane properties influence information flow within neuronal circuits is largely unknown. We studied the spatiotemporal profile of synaptic integration and short-term plasticity in dendrites of mature cerebellar stellate cells by combining two-photon guided electrical stimulation, glutamate uncaging, electron microscopy, and modeling. Synaptic activation within thin (0.4?μm) dendrites produced somatic responses that became smaller and slower with increasing distance from the soma, sublinear subthreshold input-output relationships, and a somatodendritic gradient of short-term plasticity. Unlike most studies showing that neurons employ active dendritic mechanisms, we found that passive cable properties of thin dendrites determine the sublinear integration and plasticity gradient, which both result from large?dendritic depolarizations that reduce synaptic driving force. These integrative properties allow stellate cells to act as spatiotemporal filters of synaptic input patterns, thereby biasing their output in favor of sparse presynaptic activity.     

Transfer properties of neuronal dendrites with tonically activated conductances

The somatopetal current transfer was studied in the mathematical models of a reconstructed brainstem motoneuron with tonically activated excitatory synaptic inputs uniformly distributed over dendritic arborization. The soma and axon provided a constant passive leak. The extrasynaptic dendritic membrane was either passive or active (of a Hodgkin-Huxley type). The longitudinal membrane current density (per unit path length) was used as an estimate of the current transfer effectiveness of different dendritic paths. Introduction of a steady uniform voltage-independent conductance per unit membrane area simulated such a synaptic activation. This actions always produced a spatially inhomogeneous membrane depolarization decaying from the distal dendritic tips toward the soma. The reason for such an inhomogeneity was the preponderance of somatopetal over somatofugal input conductance at every site in the dendrites with sealed distal ends and a leaky somatic end. In active dendrites, partial voltage-dependent extrasynaptic conductances followed this depolarization according to their activation-inactivation kinetics. The greater the local depolarization, the greater the contribution of the non-inactivating potassium conductance to the total membrane conductance. The contribution of the inactivated sodium conductance was one order of magnitude smaller. Correspondingly, the effective equilibrium potential of the total transmembrane current became spatially inhomogeneous and shifted to the potassium equilibrium potential. In the passive dendrites, the equilibrium potential remained spatially homogeneous. Inhomogeneities of the dendritic geometry (abrupt change in the diameter and, especially, asymmetrical branching) caused characteristic perturbations in the voltage gradient, so that the path profiles of the voltage, conductances, and currents diverged. This indicated a geometry-induced separation of the dendritic paths in their transfer effectiveness. Active dendrites of the same geometry were less effective than passive ones due to the effect of the potassium conductance associated with the hyperpolarizing equilibrium potential.     

Synaptic integration in dendritic trees

Most neurons have elaborate dendritic trees that receive tens of thousands of synaptic inputs. Because postsynaptic responses to individual synaptic events are usually small and transient, the integration of many synaptic responses is needed to depolarize most neurons to action potential threshold. Over the past decade, advances in electrical and optical recording techniques have led to new insights into how synaptic responses propagate and interact within dendritic trees. In addition to their passive electrical and morphological properties, dendrites express active conductances that shape individual synaptic responses and influence synaptic integration locally within dendrites. Dendritic voltage-gated Na(+) and Ca(2+) channels support action potential backpropagation into the dendritic tree and local initiation of dendritic spikes, whereas K(+) conductances act to dampen dendritic excitability. While all dendrites investigated to date express active conductances, different neuronal types show specific patterns of dendritic channel expression leading to cell-specific differences in the way synaptic responses are integrated within dendritic trees. This review explores the way active and passive dendritic properties shape synaptic responses in the dendrites of central neurons, and emphasizes their role in synaptic integration.     

An ephaptic transmission model of CA3 pyramidal cells: an investigation into electric field effects

Extracellular electric fields existing throughout the living brain affect the neural coding and information processing via ephaptic transmission, independent of synapses. A two-compartment whole field effect model (WFEM) of pyramidal neurons embedded within a resistive array which simulates the extracellular medium i.e. ephapse is developed to study the effects of electric field on neuronal behaviors. We derive the two linearized filed effect models (LFEM-1 and LFEM-2) from WFEM at the stable resting state. Through matching these simplified models to the subthreshold membrane response in experiments of the resting pyramidal cells exposed to applied electric fields, we not only verify our proposed model’s validity but also found the key parameters which dominate subthreshold frequency response characteristic. Moreover, we find and give its underlying biophysical mechanism that the unsymmetrical properties of active ion channels results in the very different low-frequency response of somatic and dendritic compartments. Following, WFEM is used to investigate both direct-current (DC) and alternating-current field effect on the neural firing patterns by bifurcation analyses. We present that DC electric field could modulate neuronal excitability, with the positive field improving the excitability, the modest negative field suppressing the excitability, but interestingly, the larger negative field re-exciting the neuron back into spiking behavior. The neuron exposed to the sinusoidal electric field exhibits abundant firing patterns sensitive to the input frequency and intensity. In addition, the electrical properties of ephapse can modulate the efficacy of field effect. Our simulated results are qualitatively in line with the relevant experimental results and can explain some experimental phenomena. Furthermore, they are helpful to provide the predictions which can be tested in future experiments.     

Estimation of synaptic conductances.

In order to identify and understand mechanistically the cortical circuitry of sensory information processing estimates are needed of synaptic input fields that drive neurons. From intracellular in vivo recordings one would like to estimate net synaptic conductance time courses for excitation and inhibition, g(E)(t) and g(I)(t), during time-varying stimulus presentations. However, the intrinsic conductance transients associated with neuronal spiking can confound such estimates, and thereby jeopardize functional interpretations. Here, using a conductance-based pyramidal neuron model we illustrate errors in estimates when the influence of spike-generating conductances are not reduced or avoided. A typical estimation procedure involves approximating the current-voltage relation at each time point during repeated stimuli. The repeated presentations are done in a few sets, each with a different steady bias current. From the trial-averaged smoothed membrane potential one estimates total membrane conductance and then dissects out estimates for g(E)(t) and g(I)(t). Simulations show that estimates obtained during phases without spikes are good but those obtained from phases with spiking should be viewed with skeptism. For the simulations, we consider two different synaptic input scenarios, each corresponding to computational network models of orientation tuning in visual cortex. One input scenario mimics a push-pull arrangement for g(E)(t) and g(I)(t) and idealized as specified smooth time courses. The other is taken directly from a large-scale network simulation of stochastically spiking neurons in a slab of cortex with recurrent excitation and inhibition. For both, we show that spike-generating conductances cause serious errors in the estimates of g(E) and g(I). In some phases for the push-pull examples even the polarity of g(I) is mis-estimated, indicating significant increase when g(I) is actually decreased. Our primary message is to be cautious about forming interpretations based on estimates developed during spiking phases.