Experimentally-constrained biophysical models of tonic and burst firing modes in thalamocortical neurons

A Multimodal Fitting Approach to Construct Single-Neuron Models With Patch Clamp and High-Density Microelectrode Arrays

In computational neuroscience, multicompartment models are among the most biophysically realistic representations of single neurons. Constructing such models usually involves the use of the patch-clamp technique to record somatic voltage signals under different experimental conditions. The experimental data are then used to fit the many parameters of the model. While patching of the soma is currently the gold-standard approach to build multicompartment models, several studies have also evidenced a richness of dynamics in dendritic and axonal sections. Recording from the soma alone makes it hard to observe and correctly parameterize the activity of nonsomatic compartments. In order to provide a richer set of data as input to multicompartment models, we here investigate the combination of somatic patch-clamp recordings with recordings of high-density microelectrode arrays (HD-MEAs). HD-MEAs enable the observation of extracellular potentials and neural activity of neuronal compartments at subcellular resolution. In this work, we introduce a novel framework to combine patch-clamp and HD-MEA data to construct multicompartment models. We first validate our method on a ground-truth model with known parameters and show that the use of features extracted from extracellular signals, in addition to intracellular ones, yields models enabling better fits than using intracellular features alone. We also demonstrate our procedure using experimental data by constructing cell models from in vitro cell cultures. The proposed multimodal fitting procedure has the potential to augment the modeling efforts of the computational neuroscience community and provide the field with neuronal models that are more realistic and can be better validated.

Modeling Synaptic Integration of Bursty and β Oscillatory Inputs in Ventromedial Motor Thalamic Neurons in Normal and Parkinsonian States

The ventromedial motor thalamus (VM) is implicated in multiple motor functions and occupies a central position in the cortico-basal ganglia-thalamocortical loop. It integrates glutamatergic inputs from motor cortex (MC) and motor-related subcortical areas, and it is a major recipient of inhibition from basal ganglia. Previous in vitro experiments performed in mice showed that dopamine depletion enhances the excitability of thalamocortical (TC) neurons in VM due to reduced M-type potassium currents. To understand how these excitability changes impact synaptic integration in vivo, we constructed biophysically detailed mouse VM TC model neurons fit to normal and dopamine-depleted conditions, using the NEURON simulator. These models allowed us to assess the influence of excitability changes with dopamine depletion on the integration of synaptic inputs expected in vivo We found that VM neuron models in the dopamine-depleted state showed increased firing rates with the same synaptic inputs. Synchronous bursting in inhibitory input from the substantia nigra pars reticulata (SNR), as observed in parkinsonian conditions, evoked a postinhibitory firing rate increase with a longer duration in dopamine-depleted than control conditions, due to different M-type potassium channel densities. With β oscillations in the inhibitory inputs from SNR and the excitatory inputs from cortex, we observed spike-phase locking in the activity of the models in normal and dopamine-depleted states, which relayed and amplified the oscillations of the inputs, suggesting that the increased β oscillations observed in VM of parkinsonian animals are predominantly a consequence of changes in the presynaptic activity rather than changes in intrinsic properties.

Ventral posterolateral and ventral posteromedial thalamocortical neurons have distinct physiological properties

Somatosensory information is propagated from the periphery to the cerebral cortex by two parallel pathways through the ventral posterolateral (VPL) and ventral posteromedial (VPM) thalamus. VPL and VPM neurons receive somatosensory signals from the body and head, respectively. VPL and VPM neurons may also receive cell type-specific GABAergic input from the reticular nucleus of the thalamus. Although VPL and VPM neurons have distinct connectivity and physiological roles, differences in their functional properties remain unclear as they are often studied as one ventrobasal thalamus neuron population. Here, we directly compared synaptic and intrinsic properties of VPL and VPM neurons in C57Bl/6J mice of both sexes aged P25-P32. VPL neurons showed greater depolarization-induced spike firing and spike frequency adaptation than VPM neurons. VPL and VPM neurons fired similar numbers of spikes during hyperpolarization rebound bursts, but VPM neurons exhibited shorter burst latency compared with VPL neurons, which correlated with larger sag potential. VPM neurons had larger membrane capacitance and more complex dendritic arbors. Recordings of spontaneous and evoked synaptic transmission suggested that VPL neurons receive stronger excitatory synaptic input, whereas inhibitory synapse strength was stronger in VPM neurons. This work indicates that VPL and VPM thalamocortical neurons have distinct intrinsic and synaptic properties. The observed functional differences could have important implications for their specific physiological and pathophysiological roles within the somatosensory thalamocortical network.NEW & NOTEWORTHY This study revealed that somatosensory thalamocortical neurons in the VPL and VPM have substantial differences in excitatory synaptic input and intrinsic firing properties. The distinct properties suggest that VPL and VPM neurons could process somatosensory information differently and have selective vulnerability to disease. This work improves our understanding of nucleus-specific neuron function in the thalamus and demonstrates the critical importance of studying these parallel somatosensory pathways separately.

Data-driven multiscale model of macaque auditory thalamocortical circuits reproduces in vivo dynamics

We developed a detailed model of macaque auditory thalamocortical circuits, including primary auditory cortex (A1), medial geniculate body (MGB), and thalamic reticular nucleus, utilizing the NEURON simulator and NetPyNE tool. The A1 model simulates a cortical column with over 12,000 neurons and 25 million synapses, incorporating data on cell-type-specific neuron densities, morphology, and connectivity across six cortical layers. It is reciprocally connected to the MGB thalamus, which includes interneurons and core and matrix-layer-specific projections to A1. The model simulates multiscale measures, including physiological firing rates, local field potentials (LFPs), current source densities (CSDs), and electroencephalography (EEG) signals. Laminar CSD patterns, during spontaneous activity and in response to broadband noise stimulus trains, mirror experimental findings. Physiological oscillations emerge spontaneously across frequency bands comparable to those recorded in vivo. We elucidate population-specific contributions to observed oscillation events and relate them to firing and presynaptic input patterns. The model offers a quantitative theoretical framework to integrate and interpret experimental data and predict its underlying cellular and circuit mechanisms.

A universal workflow for creation, validation, and generalization of detailed neuronal models

Detailed single-neuron modeling is widely used to study neuronal functions. While cellular and functional diversity across the mammalian cortex is vast, most of the available computational tools focus on a limited set of specific features characteristic of a single neuron. Here, we present a generalized automated workflow for the creation of robust electrical models and illustrate its performance by building cell models for the rat somatosensory cortex. Each model is based on a 3D morphological reconstruction and a set of ionic mechanisms. We use an evolutionary algorithm to optimize neuronal parameters to match the electrophysiological features extracted from experimental data. Then we validate the optimized models against additional stimuli and assess their generalizability on a population of similar morphologies. Compared to the state-of-the-art canonical models, our models show 5-fold improved generalizability. This versatile approach can be used to build robust models of any neuronal type.

Synaptic and circuit mechanisms prevent detrimentally precise correlation in the developing mammalian visual system

The developing visual thalamus and cortex extract positional information encoded in the correlated activity of retinal ganglion cells by synaptic plasticity, allowing for the refinement of connectivity. Here, we use a biophysical model of the visual thalamus during the initial visual circuit refinement period to explore the role of synaptic and circuit properties in the regulation of such neural correlations. We find that the NMDA receptor dominance, combined with weak recurrent excitation and inhibition characteristic of this age, prevents the emergence of spike-correlations between thalamocortical neurons on the millisecond timescale. Such precise correlations, which would emerge due to the broad, unrefined connections from the retina to the thalamus, reduce the spatial information contained by thalamic spikes, and therefore we term them 'parasitic' correlations. Our results suggest that developing synapses and circuits evolved mechanisms to compensate for such detrimental parasitic correlations arising from the unrefined and immature circuit.

Thalamic control of sensory processing and spindles in a biophysical somatosensory thalamoreticular circuit model of wakefulness and sleep

Thalamoreticular circuitry plays a key role in arousal, attention, cognition, and sleep spindles, and is linked to several brain disorders. A detailed computational model of mouse somatosensory thalamus and thalamic reticular nucleus has been developed to capture the properties of over 14,000 neurons connected by 6 million synapses. The model recreates the biological connectivity of these neurons, and simulations of the model reproduce multiple experimental findings in different brain states. The model shows that inhibitory rebound produces frequency-selective enhancement of thalamic responses during wakefulness. We find that thalamic interactions are responsible for the characteristic waxing and waning of spindle oscillations. In addition, we find that changes in thalamic excitability control spindle frequency and their incidence. The model is made openly available to provide a new tool for studying the function and dysfunction of the thalamoreticular circuitry in various brain states.

The role of thalamic group II mGlu receptors in health and disease

The thalamus plays a pivotal role in the integration and processing of sensory, motor, and cognitive information. It is therefore important to understand how the thalamus operates in states of both health and disease. In the present review, we discuss the function of the Group II metabotropic glutamate (mGlu) receptors within thalamic circuitry, and how they may represent therapeutic targets in treating disease states associated with thalamic dysfunction.

Large-scale biophysically detailed model of somatosensory thalamocortical circuits in NetPyNE

The primary somatosensory cortex (S1) of mammals is critically important in the perception of touch and related sensorimotor behaviors. In 2015, the Blue Brain Project (BBP) developed a groundbreaking rat S1 microcircuit simulation with over 31,000 neurons with 207 morpho-electrical neuron types, and 37 million synapses, incorporating anatomical and physiological information from a wide range of experimental studies. We have implemented this highly detailed and complex S1 model in NetPyNE, using the data available in the Neocortical Microcircuit Collaboration Portal. NetPyNE provides a Python high-level interface to NEURON and allows defining complicated multiscale models using an intuitive declarative standardized language. It also facilitates running parallel simulations, automates the optimization and exploration of parameters using supercomputers, and provides a wide range of built-in analysis functions. This will make the S1 model more accessible and simpler to scale, modify and extend in order to explore research questions or interconnect to other existing models. Despite some implementation differences, the NetPyNE model preserved the original cell morphologies, electrophysiological responses and spatial distribution for all 207 cell types; and the connectivity properties of all 1941 pathways, including synaptic dynamics and short-term plasticity (STP). The NetPyNE S1 simulations produced reasonable physiological firing rates and activity patterns across all populations. When STP was included, the network generated a 1 Hz oscillation comparable to the original model in vitro-like state. By then reducing the extracellular calcium concentration, the model reproduced the original S1 in vivo-like states with asynchronous activity. These results validate the original study using a new modeling tool. Simulated local field potentials (LFPs) exhibited realistic oscillatory patterns and features, including distance- and frequency-dependent attenuation. The model was extended by adding thalamic circuits, including 6 distinct thalamic populations with intrathalamic, thalamocortical (TC) and corticothalamic connectivity derived from experimental data. The thalamic model reproduced single known cell and circuit-level dynamics, including burst and tonic firing modes and oscillatory patterns, providing a more realistic input to cortex and enabling study of TC interactions. Overall, our work provides a widely accessible, data-driven and biophysically-detailed model of the somatosensory TC circuits that can be employed as a community tool for researchers to study neural dynamics, function and disease.

Systematic generation of biophysically detailed models with generalization capability for non-spiking neurons

Unlike spiking neurons which compress continuous inputs into digital signals for transmitting information via action potentials, non-spiking neurons modulate analog signals through graded potential responses. Such neurons have been found in a large variety of nervous tissues in both vertebrate and invertebrate species, and have been proven to play a central role in neuronal information processing. If general and vast efforts have been made for many years to model spiking neurons using conductance-based models (CBMs), very few methods have been developed for non-spiking neurons. When a CBM is built to characterize the neuron behavior, it should be endowed with generalization capabilities (i.e. the ability to predict acceptable neuronal responses to different novel stimuli not used during the model’s building). Yet, since CBMs contain a large number of parameters, they may typically suffer from a lack of such a capability. In this paper, we propose a new systematic approach based on multi-objective optimization which builds general non-spiking models with generalization capabilities. The proposed approach only requires macroscopic experimental data from which all the model parameters are simultaneously determined without compromise. Such an approach is applied on three non-spiking neurons of the nematode Caenorhabditis elegans (C. elegans), a well-known model organism in neuroscience that predominantly transmits information through non-spiking signals. These three neurons, arbitrarily labeled by convention as RIM, AIY and AFD, represent, to date, the three possible forms of non-spiking neuronal responses of C. elegans.

Cell type-specific mechanisms of information transfer in data-driven biophysical models of hippocampal CA3 principal neurons

The transformation of synaptic input into action potential output is a fundamental single-cell computation resulting from the complex interaction of distinct cellular morphology and the unique expression profile of ion channels that define the cellular phenotype. Experimental studies aimed at uncovering the mechanisms of the transfer function have led to important insights, yet are limited in scope by technical feasibility, making biophysical simulations an attractive complementary approach to push the boundaries in our understanding of cellular computation. Here we take a data-driven approach by utilizing high-resolution morphological reconstructions and patch-clamp electrophysiology data together with a multi-objective optimization algorithm to build two populations of biophysically detailed models of murine hippocampal CA3 pyramidal neurons based on the two principal cell types that comprise this region. We evaluated the performance of these models and find that our approach quantitatively matches the cell type-specific firing phenotypes and recapitulate the intrinsic population-level variability in the data. Moreover, we confirm that the conductance values found by the optimization algorithm are consistent with differentially expressed ion channel genes in single-cell transcriptomic data for the two cell types. We then use these models to investigate the cell type-specific biophysical properties involved in the generation of complex-spiking output driven by synaptic input through an information-theoretic treatment of their respective transfer functions. Our simulations identify a host of cell type-specific biophysical mechanisms that define the morpho-functional phenotype to shape the cellular transfer function and place these findings in the context of a role for bursting in CA3 recurrent network synchronization dynamics.

The hippocampus is comprised of numerous types of neurons, which constitute the cellular substrate for its rich repertoire of network dynamics. Among these are sharp waves, sequential activations of ensembles of neurons that have been shown to be crucially involved in learning and memory. In the CA3 area of the hippocampus, two types of excitatory cells, thorny and a-thorny neurons, are preferentially active during distinct phases of a sharp wave, suggesting a differential role for these cell types in phenomena such as memory consolidation. Using a strictly data-driven approach, we built biophysically realistic models of both thorny and a-thorny cells and used them to investigate the integrative differences between these two cell types. We found that both neuron classes have the capability of integrating incoming synaptic inputs in a supralinear fashion, although only a-thorny cells respond with bursts of action potentials to spatially and temporally clustered synaptic inputs. Additionally, by using a computational approach based on information theory, we show that, owing to this propensity for bursting, a-thorny cells can encode more information in their spiking output than their thorny counterpart. These results shed new light on the computational capabilities of two types of excitatory neurons and suggest that thorny and a-thorny cells may play distinct roles in the generation of hippocampal network synchronization.

Benchmarking of tools for axon length measurement in individually-labeled projection neurons

Projection neurons are the commonest neuronal type in the mammalian forebrain and their individual characterization is a crucial step to understand how neural circuitry operates. These cells have an axon whose arborizations extend over long distances, branching in complex patterns and/or in multiple brain regions. Axon length is a principal estimate of the functional impact of the neuron, as it directly correlates with the number of synapses formed by the axon in its target regions; however, its measurement by direct 3D axonal tracing is a slow and labor-intensive method. On the contrary, axon length estimations have been recently proposed as an effective and accessible alternative, allowing a fast approach to the functional significance of the single neuron. Here, we analyze the accuracy and efficiency of the most used length estimation tools—design-based stereology by virtual planes or spheres, and mathematical correction of the 2D projected-axon length—in contrast with direct measurement, to quantify individual axon length. To this end, we computationally simulated each tool, applied them over a dataset of 951 3D-reconstructed axons (from NeuroMorpho.org), and compared the generated length values with their 3D reconstruction counterparts. The evaluated reliability of each axon length estimation method was then balanced with the required human effort, experience and know-how, and economic affordability. Subsequently, computational results were contrasted with measurements performed on actual brain tissue sections. We show that the plane-based stereological method balances acceptable errors (~5%) with robustness to biases, whereas the projection-based method, despite its accuracy, is prone to inherent biases when implemented in the laboratory. This work, therefore, aims to provide a constructive benchmark to help guide the selection of the most efficient method for measuring specific axonal morphologies according to the particular circumstances of the conducted research.

Characterization of single neurons is a crucial step to understand how neural circuitry operates. Visualization of individual neurons is feasible thanks to labelling techniques that allow precise measurements at cellular resolution. This milestone gave access to powerful estimators of the functional impact of a neuron, such as axon length. Although techniques relying on direct 3D reconstruction of individual axons are the gold standard, handiness and accessibility are still an issue. Indirect estimations of axon length have been proposed as agile and effective alternatives, each offering different solutions to the accuracy-cost tradeoff. In this work we report a computational benchmarking between three experimental tools used for axon length estimation on brain tissue sections. Performance of each tool was simulated and tested for 951 3D-reconstructed axons, by comparing estimated axon lengths against direct measurements. Assessment of suitability to different research and funding circumstances is also provided, taking into consideration factors such as training expertise, economic cost and required equipment, alongside methodological results. These findings could be an important reference for research on neuronal wiring, as well as for broader studies involving neuroanatomical and neural circuit modelling.

Corticothalamic Pathways From Layer 5: Emerging Roles in Computation and Pathology

Large portions of the thalamus receive strong driving input from cortical layer 5 (L5) neurons but the role of this important pathway in cortical and thalamic computations is not well understood. L5-recipient "higher-order" thalamic regions participate in cortico-thalamo-cortical (CTC) circuits that are increasingly recognized to be (1) anatomically and functionally distinct from better-studied "first-order" CTC networks, and (2) integral to cortical activity related to learning and perception. Additionally, studies are beginning to elucidate the clinical relevance of these networks, as dysfunction across these pathways have been implicated in several pathological states. In this review, we highlight recent advances in understanding L5 CTC networks across sensory modalities and brain regions, particularly studies leveraging cell-type-specific tools that allow precise experimental access to L5 CTC circuits. We aim to provide a focused and accessible summary of the anatomical, physiological, and computational properties of L5-originating CTC networks, and outline their underappreciated contribution in pathology. We particularly seek to connect single-neuron and synaptic properties to network (dys)function and emerging theories of cortical computation, and highlight information processing in L5 CTC networks as a promising focus for computational studies.

Computational Evidence for a Competitive Thalamocortical Model of Spikes and Spindle Activity in Rolandic Epilepsy

Rolandic epilepsy (RE) is the most common idiopathic focal childhood epilepsy syndrome, characterized by sleep-activated epileptiform spikes and seizures and cognitive deficits in school age children. Recent evidence suggests that this disease may be caused by disruptions to the Rolandic thalamocortical circuit, resulting in both an abundance of epileptiform spikes and a paucity of sleep spindles in the Rolandic cortex during non-rapid eye movement sleep (NREM); electrographic features linked to seizures and cognitive symptoms, respectively. The neuronal mechanisms that support the competitive shared thalamocortical circuitry between pathological epileptiform spikes and physiological sleep spindles are not well-understood. In this study we introduce a computational thalamocortical model for the sleep-activated epileptiform spikes observed in RE. The cellular and neuronal circuits of this model incorporate recent experimental observations in RE, and replicate the electrophysiological features of RE. Using this model, we demonstrate that: (1) epileptiform spikes can be triggered and promoted by either a reduced NMDA current or h-type current; and (2) changes in inhibitory transmission in the thalamic reticular nucleus mediates an antagonistic dynamic between epileptiform spikes and spindles. This work provides the first computational model that both recapitulates electrophysiological features and provides a mechanistic explanation for the thalamocortical switch between the pathological and physiological electrophysiological rhythms observed during NREM sleep in this common epileptic encephalopathy.

A new reduced-morphology model for CA1 pyramidal cells and its validation and comparison with other models using HippoUnit

Modeling long-term neuronal dynamics may require running long-lasting simulations. Such simulations are computationally expensive, and therefore it is advantageous to use simplified models that sufficiently reproduce the real neuronal properties. Reducing the complexity of the neuronal dendritic tree is one option. Therefore, we have developed a new reduced-morphology model of the rat CA1 pyramidal cell which retains major dendritic branch classes. To validate our model with experimental data, we used HippoUnit, a recently established standardized test suite for CA1 pyramidal cell models. The HippoUnit allowed us to systematically evaluate the somatic and dendritic properties of the model and compare them to models publicly available in the ModelDB database. Our model reproduced (1) somatic spiking properties, (2) somatic depolarization block, (3) EPSP attenuation, (4) action potential backpropagation, and (5) synaptic integration at oblique dendrites of CA1 neurons. The overall performance of the model in these tests achieved higher biological accuracy compared to other tested models. We conclude that, due to its realistic biophysics and low morphological complexity, our model captures key physiological features of CA1 pyramidal neurons and shortens computational time, respectively. Thus, the validated reduced-morphology model can be used for computationally demanding simulations as a substitute for more complex models.

Rodent somatosensory thalamocortical circuitry: Neurons, synapses, and connectivity

As our understanding of the thalamocortical system deepens, the questions we face become more complex. Their investigation requires the adoption of novel experimental approaches complemented with increasingly sophisticated computational modeling. In this review, we take stock of current data and knowledge about the circuitry of the somatosensory thalamocortical loop in rodents, discussing common principles across modalities and species whenever appropriate. We review the different levels of organization, including the cells, synapses, neuroanatomy, and network connectivity. We provide a complete overview of this system that should be accessible for newcomers to this field while nevertheless being comprehensive enough to serve as a reference for seasoned neuroscientists and computational modelers studying the thalamocortical system. We further highlight key gaps in data and knowledge that constitute pressing targets for future experimental work. Filling these gaps would provide invaluable information for systematically unveiling how this system supports behavioral and cognitive processes.

Changes in Excitability Properties of Ventromedial Motor Thalamic Neurons in 6-OHDA Lesioned Mice

The activity of basal ganglia input receiving motor thalamus (BGMT) makes a critical impact on motor cortical processing, but modification in BGMT processing with Parkinsonian conditions has not be investigated at the cellular level. Such changes may well be expected because of homeostatic regulation of neural excitability in the presence of altered synaptic drive with dopamine depletion. We addressed this question by comparing BGMT properties in brain slice recordings between control and unilaterally 6-hydroxydopamine hydrochloride (6-OHDA)-treated adult mice. At a minimum of one month after 6-OHDA treatment, BGMT neurons showed a highly significant increase in intrinsic excitability, which was primarily because of a decrease in M-type potassium current. BGMT neurons after 6-OHDA treatment also showed an increase in T-type calcium rebound spikes following hyperpolarizing current steps. Biophysical computer modeling of a thalamic neuron demonstrated that an increase in rebound spiking can also be accounted for by a decrease in the M-type potassium current. Modeling also showed that an increase in sag with hyperpolarizing steps found after 6-OHDA treatment could in part but not fully be accounted for by the decrease in M-type current. These findings support the hypothesis that homeostatic changes in BGMT neural properties following 6-OHDA treatment likely influence the signal processing taking place in the BG thalamocortical network in Parkinson’s disease.

On the Modeling of the Three Types of Non-spiking Neurons of the Caenorhabditis elegans

The nematode Caenorhabditis elegans (C. elegans) is a well-known model organism in neuroscience. The relative simplicity of its nervous system, made up of few hundred neurons, shares some essential features with more sophisticated nervous systems, including the human one. If we are able to fully characterize the nervous system of this organism, we will be one step closer to understanding the mechanisms underlying the behavior of living things. Following a recently conducted electrophysiological survey on different C. elegans neurons, this paper aims at modeling the three non-spiking RIM, AIY and AFD neurons (arbitrarily named with three upper case letters by convention). To date, they represent the three possible forms of non-spiking neuronal responses of the C. elegans. To achieve this objective, we propose a conductance-based neuron model adapted to the electrophysiological features of each neuron. These features are based on current biological research and a series of in-silico experiments which use differential evolution to fit the model to experimental data. From the obtained results, we formulate a series of biological hypotheses regarding currents involved in the neuron dynamics. These models reproduce experimental data with a high degree of accuracy while being biologically consistent with state-of-the-art research.

An efficient analytical reduction of detailed nonlinear neuron models

Detailed conductance-based nonlinear neuron models consisting of thousands of synapses are key for understanding of the computational properties of single neurons and large neuronal networks, and for interpreting experimental results. Simulations of these models are computationally expensive, considerably curtailing their utility. Neuron_Reduce is a new analytical approach to reduce the morphological complexity and computational time of nonlinear neuron models. Synapses and active membrane channels are mapped to the reduced model preserving their transfer impedance to the soma; synapses with identical transfer impedance are merged into one NEURON process still retaining their individual activation times. Neuron_Reduce accelerates the simulations by 40-250 folds for a variety of cell types and realistic number (10,000-100,000) of synapses while closely replicating voltage dynamics and specific dendritic computations. The reduced neuron-models will enable realistic simulations of neural networks at unprecedented scale, including networks emerging from micro-connectomics efforts and biologically-inspired "deep networks". Neuron_Reduce is publicly available and is straightforward to implement.