A reduced compartmental model of the mitral cell for use in network models of the olfactory bulb

Data-driven reduction of dendritic morphologies with preserved dendro-somatic responses

Dendrites shape information flow in neurons. Yet, there is little consensus on the level of spatial complexity at which they operate. Through carefully chosen parameter fits, solvable in the least-squares sense, we obtain accurate reduced compartmental models at any level of complexity. We show that (back-propagating) action potentials, Ca²⁺ spikes, and N-methyl-D-aspartate spikes can all be reproduced with few compartments. We also investigate whether afferent spatial connectivity motifs admit simplification by ablating targeted branches and grouping affected synapses onto the next proximal dendrite. We find that voltage in the remaining branches is reproduced if temporal conductance fluctuations stay below a limit that depends on the average difference in input resistance between the ablated branches and the next proximal dendrite. Furthermore, our methodology fits reduced models directly from experimental data, without requiring morphological reconstructions. We provide software that automatizes the simplification, eliminating a common hurdle toward including dendritic computations in network models.

Generative Biophysical Modeling of Dynamical Networks in the Olfactory System

Generative models are computational models designed to generate appropriate values for all of their embedded variables, thereby simulating the response properties of a complex system based on the coordinated interactions of a multitude of physical mechanisms. In systems neuroscience, generative models are generally biophysically based compartmental models of neurons and networks that are explicitly multiscale, being constrained by experimental data at multiple levels of organization from cellular membrane properties to large-scale network dynamics. As such, they are able to explain the origins of emergent properties in complex systems, and serve as tests of sufficiency and as quantitative instantiations of working hypotheses that may be too complex to simply intuit. Moreover, when adequately constrained, generative biophysical models are able to predict novel experimental outcomes, and consequently are powerful tools for experimental design. We here outline a general strategy for the iterative design and implementation of generative, multiscale biophysical models of neural systems. We illustrate this process using our ongoing, iteratively developing model of the mammalian olfactory bulb. Because the olfactory bulb exhibits diverse and interesting properties at multiple scales of organization, it is an attractive system in which to illustrate the value of generative modeling across scales.

Intraglomerular gap junctions enhance interglomerular synchrony in a sparsely connected olfactory bulb network

KEY POINTS

Despite sparse connectivity, population-level interactions between mitral cells (MCs) and granule cells (GCs) can generate synchronized oscillations in the rodent olfactory bulb. Intraglomerular gap junctions between MCs at the same glomerulus can greatly enhance synchronized activity of MCs at different glomeruli. The facilitating effect of intraglomerular gap junctions on interglomerular synchrony is through triggering of mutually synchronizing interactions between MCs and GCs. Divergent connections between MCs and GCs make minimal direct contribution to synchronous activity.

ABSTRACT

A dominant feature of the olfactory bulb response to odour is fast synchronized oscillations at beta (15-40 Hz) or gamma (40-90 Hz) frequencies, thought to be involved in integration of olfactory signals. Mechanistically, the bulb presents an interesting case study for understanding how beta/gamma oscillations arise. Fast oscillatory synchrony in the activity of output mitral cells (MCs) appears to result from interactions with GABAergic granule cells (GCs), yet the incidence of MC-GC connections is very low, around 4%. Here, we combined computational and experimental approaches to examine how oscillatory synchrony can nevertheless arise, focusing mainly on activity between 'non-sister' MCs affiliated with different glomeruli (interglomerular synchrony). In a sparsely connected model of MCs and GCs, we found first that interglomerular synchrony was generally quite low, but could be increased by a factor of 4 by physiological levels of gap junctional coupling between sister MCs at the same glomerulus. This effect was due to enhanced mutually synchronizing interactions between MC and GC populations. The potent role of gap junctions was confirmed in patch-clamp recordings in bulb slices from wild-type and connexin 36-knockout (KO) mice. KO reduced both beta and gamma local field potential oscillations as well as synchrony of inhibitory signals in pairs of non-sister MCs. These effects were independent of potential KO actions on network excitation. Divergent synaptic connections did not contribute directly to the vast majority of synchronized signals. Thus, in a sparsely connected network, gap junctions between a small subset of cells can, through population effects, greatly amplify oscillatory synchrony amongst unconnected cells.

Distinct current modules shape cellular dynamics in model neurons

Numerous intrinsic currents are known to collectively shape neuronal membrane potential dynamics, or neuronal signatures. Although how sets of currents shape specific signatures such as spiking characteristics or oscillations has been studied individually, it is less clear how a neuron's suite of currents jointly shape its entire set of signatures. Biophysical conductance-based models of neurons represent a viable tool to address this important question. We hypothesized that currents are grouped into distinct modules that shape specific neuronal characteristics or signatures, such as resting potential, sub-threshold oscillations, and spiking waveforms, for several classes of neurons. For such a grouping to occur, the currents within one module should have minimal functional interference with currents belonging to other modules. This condition is satisfied if the gating functions of currents in the same module are grouped together on the voltage axis; in contrast, such functions are segregated along the voltage axis for currents belonging to different modules. We tested this hypothesis using four published example case models and found it to be valid for these classes of neurons. This insight into the neurobiological organization of currents also suggests an intuitive, systematic, and robust methodology to develop biophysical single-cell models with multiple biological characteristics applicable for both hand- and automated-tuning approaches. We illustrate the methodology using two example case rodent pyramidal neurons, from the lateral amygdala and the hippocampus. The methodology also helped reveal that a single-core compartment model could capture multiple neuronal properties. Such biophysical single-compartment models have potential to improve the fidelity of large network models.

Using Strahler's analysis to reduce up to 200-fold the run time of realistic neuron models

The cellular mechanisms underlying higher brain functions/dysfunctions are extremely difficult to investigate experimentally, and detailed neuron models have proven to be a very useful tool to help these kind of investigations. However, realistic neuronal networks of sizes appropriate to study brain functions present the major problem of requiring a prohibitively high computational resources. Here, building on our previous work, we present a general reduction method based on Strahler's analysis of neuron morphologies. We show that, without any fitting or tuning procedures, it is possible to map any morphologically and biophysically accurate neuron model into an equivalent reduced version. Using this method for Purkinje cells, we demonstrate how run times can be reduced up to 200–fold, while accurately taking into account the effects of arbitrarily located and activated synaptic inputs.

A two-layer biophysical model of cholinergic neuromodulation in olfactory bulb

Cholinergic inputs from the basal forebrain regulate multiple olfactory bulb (OB) functions, including odor discrimination, perceptual learning, and short-term memory. Previous studies have shown that nicotinic cholinergic receptor activation sharpens mitral cell chemoreceptive fields, likely via intraglomerular circuitry. Muscarinic cholinergic activation is less well understood, though muscarinic receptors are implicated in olfactory learning and in the regulation of synchronized oscillatory dynamics in hippocampus and cortex. To understand the mechanisms underlying cholinergic neuromodulation in OB, we developed a biophysical model of the OB neuronal network including both glomerular layer and external plexiform layer (EPL) computations and incorporating both nicotinic and muscarinic neuromodulatory effects. Our simulations show how nicotinic activation within glomerular circuits sharpens mitral cell chemoreceptive fields, even in the absence of EPL circuitry, but does not facilitate intrinsic oscillations or spike synchronization. In contrast, muscarinic receptor activation increases mitral cell spike synchronization and field oscillatory power by potentiating granule cell excitability and lateral inhibitory interactions within the EPL, but it has little effect on mitral cell firing rates and hence does not sharpen olfactory representations under a rate metric. These results are consistent with the theory that EPL interactions regulate the timing, rather than the existence, of mitral cell action potentials and perform their computations with respect to a spike timing-based metric. This general model suggests that the roles of nicotinic and muscarinic receptors in olfactory bulb are both distinct and complementary to one another, together regulating the effects of ascending cholinergic inputs on olfactory bulb transformations.

The periglomerular cell of the olfactory bulb and its role in controlling mitral cell spiking: a computational model

Interneurons in the olfactory bulb are key elements of odor processing but their roles have not yet being fully understood. Two types of inhibitory interneurons, periglomerular and granule cells, act at two different levels within the olfactory bulb and may have different roles in coordinating the spiking of mitral cells, which are the principal output neurons of the olfactory bulb. In this work we introduce a reduced compartmental model of the periglomerular cell and use it to investigate its role on mitral cell spiking in a model of an elementary cell triad composed of these two cell types plus a granule cell. Our simulation results show that the periglomerular cell is more effective in inhibiting the mitral cell than the granule cell. Based on our results we predict that periglomerular and granule cells have different roles in the control of mitral cell spiking. The periglomerular cell would be the only one capable of completely inhibiting the mitral cell, and the activity decrease of the mitral cell through this inhibitory action would occur in a stepwise fashion depending on parameters of the periglomerular and granule cells as well as on the relative times of arrival of external stimuli to the three cells. The major role of the granule cell would be to facilitate the inhibitory action of the periglomerular cell by enlarging the range of parameters of the periglomerular cell which correspond to complete inhibition of the mitral cell. The combined action of the two interneurons would thus provide an efficient way of controling the instantaneous value of the firing rate of the mitral cell.

Progress in defining heterogeneity and modeling periglomerular cells in the olfactory bulb

In recent years the evolution of olfactory bulb periglomerular cells, as well as the function of periglomerular cells in olfactory encoding, has attracted increasing attention. Studies of neural information encoding based on the analysis of simulation and modeling have given rise to electrophysiological models of periglomerular cells, which have an important role in the understanding of the biology of these cells. In this review we provide a brief introduction to the anatomy of the olfactory system and the cell types in the olfactory bulb. We elaborate on the latest progress in the study of the heterogeneity of periglomerular cells based on different classification criteria, such as molecular markers, structure, ion channels and action potentials. Then, we discuss the several existing electrophysiological models of periglomerular cells, and we highlight the problems and defects of these models. Finally, considering our present work, we propose a future direction for electrophysiological investigations of periglomerular cells and for the modeling of periglomerular cells and olfactory information encoding.

The use of automated parameter searches to improve ion channel kinetics for neural modeling

The voltage and time dependence of ion channels can be regulated, notably by phosphorylation, interaction with phospholipids, and binding to auxiliary subunits. Many parameter variation studies have set conductance densities free while leaving kinetic channel properties fixed as the experimental constraints on the latter are usually better than on the former. Because individual cells can tightly regulate their ion channel properties, we suggest that kinetic parameters may be profitably set free during model optimization in order to both improve matches to data and refine kinetic parameters. To this end, we analyzed the parameter optimization of reduced models of three electrophysiologically characterized and morphologically reconstructed globus pallidus neurons. We performed two automated searches with different types of free parameters. First, conductance density parameters were set free. Even the best resulting models exhibited unavoidable problems which were due to limitations in our channel kinetics. We next set channel kinetics free for the optimized density matches and obtained significantly improved model performance. Some kinetic parameters consistently shifted to similar new values in multiple runs across three models, suggesting the possibility for tailored improvements to channel models. These results suggest that optimized channel kinetics can improve model matches to experimental voltage traces, particularly for channels characterized under different experimental conditions than recorded data to be matched by a model. The resulting shifts in channel kinetics from the original template provide valuable guidance for future experimental efforts to determine the detailed kinetics of channel isoforms and possible modulated states in particular types of neurons.

The capabilities and limitations of conductance-based compartmental neuron models with reduced branched or unbranched morphologies and active dendrites

Conductance-based neuron models are frequently employed to study the dynamics of biological neural networks. For speed and ease of use, these models are often reduced in morphological complexity. Simplified dendritic branching structures may process inputs differently than full branching structures, however, and could thereby fail to reproduce important aspects of biological neural processing. It is not yet well understood which processing capabilities require detailed branching structures. Therefore, we analyzed the processing capabilities of full or partially branched reduced models. These models were created by collapsing the dendritic tree of a full morphological model of a globus pallidus (GP) neuron while preserving its total surface area and electrotonic length, as well as its passive and active parameters. Dendritic trees were either collapsed into single cables (unbranched models) or the full complement of branch points was preserved (branched models). Both reduction strategies allowed us to compare dynamics between all models using the same channel density settings. Full model responses to somatic inputs were generally preserved by both types of reduced model while dendritic input responses could be more closely preserved by branched than unbranched reduced models. However, features strongly influenced by local dendritic input resistance, such as active dendritic sodium spike generation and propagation, could not be accurately reproduced by any reduced model. Based on our analyses, we suggest that there are intrinsic differences in processing capabilities between unbranched and branched models. We also indicate suitable applications for different levels of reduction, including fast searches of full model parameter space.

Specific entrainment of mitral cells during gamma oscillation in the rat olfactory bulb

Local field potential (LFP) oscillations are often accompanied by synchronization of activity within a widespread cerebral area. Thus, the LFP and neuronal coherence appear to be the result of a common mechanism that underlies neuronal assembly formation. We used the olfactory bulb as a model to investigate: (1) the extent to which unitary dynamics and LFP oscillations can be correlated and (2) the precision with which a model of the hypothesized underlying mechanisms can accurately explain the experimental data. For this purpose, we analyzed simultaneous recordings of mitral cell (MC) activity and LFPs in anesthetized and freely breathing rats in response to odorant stimulation. Spike trains were found to be phase-locked to the gamma oscillation at specific firing rates and to form odor-specific temporal patterns. The use of a conductance-based MC model driven by an approximately balanced excitatory-inhibitory input conductance and a relatively small inhibitory conductance that oscillated at the gamma frequency allowed us to provide one explanation of the experimental data via a mode-locking mechanism. This work sheds light on the way network and intrinsic MC properties participate in the locking of MCs to the gamma oscillation in a realistic physiological context and may result in a particular time-locked assembly. Finally, we discuss how a self-synchronization process with such entrainment properties can explain, under experimental conditions: (1) why the gamma bursts emerge transiently with a maximal amplitude position relative to the stimulus time course; (2) why the oscillations are prominent at a specific gamma frequency; and (3) why the oscillation amplitude depends on specific stimulus properties. We also discuss information processing and functional consequences derived from this mechanism.

Olfactory function relies on a chain of neural relays that extends from the periphery to the central nervous system and implies neural activity with various timescales. A central question in neuroscience is how information is encoded by the neural activity. In the mammalian olfactory bulb, local neural activity oscillations in the 40–80 Hz range (gamma) may influence the timing of individual neuron activities such that olfactory information may be encoded in this way. In this study, we first characterize in vivo the detailed activity of individual neurons relative to the oscillation and find that, depending on their state, neurons can exhibit periodic activity patterns. We also find, at least qualitatively, a relation between this activity and a particular odor. This is reminiscent of general physical phenomena—the entrainment by an oscillation—and to verify this hypothesis, in a second phase, we build a biologically realistic model mimicking these in vivo conditions. Our model confirms quantitatively this hypothesis and reveals that entrainment is maximal in the gamma range. Taken together, our results suggest that the neuronal activity may be specifically formatted in time during the gamma oscillation in such a way that it could, at this stage, encode the odor.

Automated neuron model optimization techniques: a review

The increase in complexity of computational neuron models makes the hand tuning of model parameters more difficult than ever. Fortunately, the parallel increase in computer power allows scientists to automate this tuning. Optimization algorithms need two essential components. The first one is a function that measures the difference between the output of the model with a given set of parameter and the data. This error function or fitness function makes the ranking of different parameter sets possible. The second component is a search algorithm that explores the parameter space to find the best parameter set in a minimal amount of time. In this review we distinguish three types of error functions: feature-based ones, point-by-point comparison of voltage traces and multi-objective functions. We then detail several popular search algorithms, including brute-force methods, simulated annealing, genetic algorithms, evolution strategies, differential evolution and particle-swarm optimization. Last, we shortly describe Neurofitter, a free software package that combines a phase-plane trajectory density fitness function with several search algorithms.

Lateral dendritic shunt inhibition can regularize mitral cell spike patterning

Mitral cells, the principal output neurons of the olfactory bulb, receive direct synaptic activation from primary sensory neurons. Shunting inhibitory inputs delivered by granule cell interneurons onto mitral cell lateral dendrites, while poorly positioned to prevent spike initiation, are believed to influence spike timing and underlie coordinated field potential oscillations. We investigated this phenomenon in a reduced compartmental mitral cell model suitable for incorporation into network simulations. Lateral dendritic shunt conductances delayed spiking to a degree dependent on both their electrotonic distance and phase of onset. Moreover, when the afferent activation of mitral cells was loosely coordinated in time, recurrent inhibition significantly narrowed the distribution of mitral cell spike times, illustrating a tendency towards coordinated synchronous activity. However, if mitral cell activity was initially disorganized, recurrent inhibition actually increased the variance in spike timing. This result suggests an essential role for early mechanisms of temporal coordination in olfaction, such as sniffing and the initial synchronization of mitral cell intrinsic oscillations by periglomerular cell-mediated inhibition.

Neurofitter: a parameter tuning package for a wide range of electrophysiological neuron models

The increase in available computational power and the higher quality of experimental recordings have turned the tuning of neuron model parameters into a problem that can be solved by automatic global optimization algorithms. Neurofitter is a software tool that interfaces existing neural simulation software and sophisticated optimization algorithms with a new way to compute the error measure. This error measure represents how well a given parameter set is able to reproduce the experimental data. It is based on the phase-plane trajectory density method, which is insensitive to small phase differences between model and data. Neurofitter enables the effortless combination of many different time-dependent data traces into the error measure, allowing the neuroscientist to focus on what are the seminal properties of the model.We show results obtained by applying Neurofitter to a simple single compartmental model and a complex multi-compartmental Purkinje cell (PC) model. These examples show that the method is able to solve a variety of tuning problems and demonstrate details of its practical application.

Creation and reduction of a morphologically detailed model of a leech heart interneuron

Conductance-based neuron models aid in understanding the role intrinsic and synaptic currents play in producing neuronal activity. Incorporating morphological detail into a model allows for additional analysis of nonhomogeneous distributions of active and synaptic conductances, as well as spatial segregation of electrical events. We developed a morphologically detailed "Full Model" of a leech heart interneuron that replicates reasonably well intracellular recordings from these interneurons. However, it constitutes hundreds of compartments, each increasing parameter space and simulation time. To reduce the number of compartments of the Full Model, while preserving conductance densities and distributions, its compartments were grouped into functional groups that each share identical conductance densities. Each functional group was sequentially reduced to one or two compartments, preserving surface area, conductance densities, and its contribution to input resistance. As a result, the input resistance and membrane time constant were preserved. The axial resistances of several compartments were rescaled to match the amplitude of synaptic currents and low-threshold calcium currents and the shape of action potentials to those in the Full Model. This reduced model, with intrinsic conductances, matched the activity of the Full Model for a variety of simulated current-clamp and voltage-clamp data. Because surface area and conductance distribution of the functional groups of the Full Model were maintained, parameter changes introduced into the reduced model can be directly translated to the Full Model. Thus our computationally efficient reduced morphology model can be used as a tool for exploring the parameter space of the Full Model and in network simulations.

Dynamical mechanisms of odor processing in olfactory bulb mitral cells

In the olfactory system, the contribution of dynamical properties such as neuronal oscillations and spike synchronization to the representation of odor stimuli is a matter of substantial debate. While relatively simple computational models have sufficed to guide current research in large-scale network dynamics, less attention has been paid to modeling the membrane dynamics in bulbar neurons that may be equally essential to sensory processing. We here present a reduced, conductance-based compartmental model of olfactory bulb mitral cells that exhibits the complex dynamical properties observed in these neurons. Specifically, model neurons exhibit intrinsic subthreshold oscillations with voltage-dependent frequencies that shape the timing of stimulus-evoked action potentials. These oscillations rely on a persistent sodium conductance, an inactivating potassium conductance, and a calcium-dependent potassium conductance and are reset via inhibitory input such as that delivered by periglomerular cell shunt inhibition. Mitral cells fire bursts, or clusters, of spikes when continuously stimulated. Burst properties depend critically on multiple currents, but a progressive deinactivation of I(A) over the course of a burst is an important regulator of burst termination. Each of these complex properties exhibits appropriate dynamics and pharmacology as determined by electrophysiological studies. Additionally, we propose that a second, inconsistently observed form of infrathreshold bistability in mitral cells may derive from the activation of ATP-activated potassium currents responding to hypoxic conditions. We discuss the integration of these cellular properties in the larger context of olfactory bulb network operations.

Circuit Properties Generating Gamma Oscillations in a Network Model of the Olfactory Bulb

The study of the neural basis of olfaction is important both for understanding the sense of smell and for understanding the mechanisms of neural computation. In the olfactory bulb (OB), the spatial patterning of both sensory inputs and synaptic interactions is crucial for processing odor information, although this patterning alone is not sufficient. Recent studies have suggested that representations of odor may already be distributed and dynamic in the first olfactory relay. The growing evidence demonstrating a functional role for the temporal structure of bulbar neuronal activity supports this assumption. However, the detailed mechanisms underlying this temporal structure have never been thoroughly studied. Our study focused on gamma (40–100 Hz) network oscillations in the mammalian OB, which is a form of temporal patterning in bulbar activity elicited by olfactory stimuli. We used computational modeling combined with electrophysiological recordings to investigate the basic synaptic organization necessary and sufficient to generate sustained gamma rhythms. We found that features of gamma oscillations obtained in vitro were identical to those of a model based on lateral inhibition as the coupling modality (i.e., low irregular firing rate and high oscillation stability). In contrast, they differed substantially from those of a model based on lateral excitatory coupling (i.e., high regular firing rate and instable oscillations). Therefore we could precisely tune the oscillation frequency by changing the kinetics of inhibitory events supporting the lateral inhibition. Moreover, gradually decreasing GABAergic synaptic transmission decreased the degree of relay neuron synchronization in response to sensory inputs, both theoretically and experimentally. Thus we have shown that lateral inhibition provides a mechanism by which the dynamic processing of odor information might be finely tuned within the OB circuit.

Computation in the olfactory system

Computational models are increasingly essential to systems neuroscience. Models serve as proofs of concept, tests of sufficiency, and as quantitative embodiments of working hypotheses and are important tools for understanding and interpreting complex data sets. In the olfactory system, models have played a particularly prominent role in framing contemporary theories and presenting novel hypotheses, a role that will only grow as the complexity and intricacy of experimental data continue to increase. This review will attempt to provide a comprehensive, functional overview of computational ideas in olfaction and outline a computational framework for olfactory processing based on the insights provided by these diverse models and their supporting data.

Dendrodendritic inhibition and simulated odor responses in a detailed olfactory bulb network model

In the olfactory bulb, both the spatial distribution and the temporal structure of neuronal activity appear to be important for processing odor information, but it is currently impossible to measure both of these simultaneously with high resolution and in all layers of the bulb. We have developed a biologically realistic model of the mammalian olfactory bulb, incorporating the mitral and granule cells and the dendrodendritic synapses between them, which allows us to observe the network behavior in detail. The cell models were based on previously published work. The attributes of the synapses were obtained from the literature. The pattern of synaptic connections was based on the limited experimental data in the literature on the statistics of connections between neurons in the bulb. The results of simulation experiments with electrical stimulation agree closely in most details with published experimental data. This gives confidence that the model is capturing features of network interactions in the real olfactory bulb. The model predicts that the time course of dendrodendritic inhibition is dependent on the network connectivity as well as on the intrinsic parameters of the synapses. In response to simulated odor stimulation, strongly activated mitral cells tend to suppress neighboring cells, the mitral cells readily synchronize their firing, and increasing the stimulus intensity increases the degree of synchronization. Preliminary experiments suggest that slow temporal changes in the degree of synchronization are more useful in distinguishing between very similar odorants than is the spatial distribution of mean firing rate.