Phase-Dependent Response to Electrical Stimulation of Cortical Networks during Recurrent Epileptiform Short Discharge Generation In Vitro.

The closed-loop control of pathological brain activity is a challenging task. In this study, we investigated the sensitivity of continuous epileptiform short discharge generation to electrical stimulation applied at different phases between the discharges using an in vitro 4-AP-based model of epilepsy in rat hippocampal slices. As a measure of stimulation effectiveness, we introduced a sensitivity function, which we then measured in experiments and analyzed with different biophysical and abstract mathematical models, namely, (i) the two-order subsystem of our previous Epileptor-2 model, describing short discharge generation governed by synaptic resource dynamics; (ii) a similar model governed by shunting conductance dynamics (Epileptor-2B); (iii) the stochastic leaky integrate-and-fire (LIF)-like model applied for the network; (iv) the LIF model with potassium M-channels (LIF+KM), belonging to Class II of excitability; and (v) the Epileptor-2B model with after-spike depolarization. A semi-analytic method was proposed for calculating the interspike interval (ISI) distribution and the sensitivity function in LIF and LIF+KM models, which provided parametric analysis. Sensitivity was found to increase with phase for all models except the last one. The Epileptor-2B model is favored over other models for subthreshold oscillations in the presence of large noise, based on the comparison of ISI statistics and sensitivity functions with experimental data. This study also emphasizes the stochastic nature of epileptiform discharge generation and the greater effectiveness of closed-loop stimulation in later phases of ISIs.

Ictal wavefront propagation in slices and simulations with conductance-based refractory density model.

The mechanisms determining ictal discharge (ID) propagation are still not clear. In the present study, we aimed to examine these mechanisms in animal and mathematical models of epileptiform activity. Using double-patch and extracellular potassium ion concentration recordings in rat hippocampal-cortical slices, we observed that IDs moved at a speed of about 1 mm/s or less. The mechanisms of such slow propagation have been studied with a mathematical, conductance-based refractory density (CBRD) model that describes the GABA- and glutamatergic neuronal populations' interactions and ion dynamics in brain tissue. The modeling study reveals two main factors triggerring IDs: (i) increased interneuronal activity leading to chloride ion accumulation and a consequent depolarizing GABAergic effect and (ii) the elevation of extracellular potassium ion concentration. The local synaptic transmission followed by local potassium ion extrusion and GABA receptor-mediated chloride ion accumulation underlies the ID wavefront's propagation. In contrast, potassium ion diffusion in the extracellular space is slower and does not affect ID's speed. The short discharges, constituting the ID, propagate much faster than the ID front. The accumulation of sodium ions inside neurons due to their hyperactivity and glutamatergic currents boosts the Na+/K+ pump, which terminates the ID. Knowledge of the mechanism of ID generation and propagation contributes to the development of new treatments against epilepsy.

Single-compartment model of a pyramidal neuron, fitted to recordings with current and conductance injection.

For single neuron models, reproducing characteristics of neuronal activity such as the firing rate, amplitude of spikes, and threshold potentials as functions of both synaptic current and conductance is a challenging task. In the present work, we measure these characteristics of regular spiking cortical neurons using the dynamic patch-clamp technique, compare the data with predictions from the standard Hodgkin-Huxley and Izhikevich models, and propose a relatively simple five-dimensional dynamical system model, based on threshold criteria. The model contains a single sodium channel with slow inactivation, fast activation and moderate deactivation, as well as, two fast repolarizing and slow shunting potassium channels. The model quantitatively reproduces characteristics of steady-state activity that are typical for a cortical pyramidal neuron, namely firing rate not exceeding 30 Hz; critical values of the stimulating current and conductance which induce the depolarization block not exceeding 80 mV and 3, respectively (both values are scaled by the resting input conductance); extremum of hyperpolarization close to the midpoint between spikes. The analysis of the model reveals that the spiking regime appears through a saddle-node-on-invariant-circle bifurcation, and the depolarization block is reached through a saddle-node bifurcation of cycles. The model can be used for realistic network simulations, and it can also be implemented within the so-called mean-field, refractory density framework.

Hyperbolic equations for neuronal membrane deformation waves accompanying an action potential.

Experiments show that the propagation of an action potential along an axon is accompanied by mechanical deformations. We describe the mechanisms of the effect using fluid dynamic equations, Laplace's and Hook's laws for surface tension, and Lippmann's law, which relates membrane tension to membrane potential. We derived a minimal, 1-D model, which is a hyperbolic system of equations. Our model qualitatively reproduces the membrane's mechanical deformation evoked by either the propagation of an action potential or the stepwise change of membrane potential. The understanding of the relationship between electrical activity and mechanical deformation provides guidance toward non-invasive imaging of neuronal activity.

A strategy for mapping biophysical to abstract neuronal network models applied to primary visual cortex.

A fundamental challenge for the theoretical study of neuronal networks is to make the link between complex biophysical models based directly on experimental data, to progressively simpler mathematical models that allow the derivation of general operating principles. We present a strategy that successively maps a relatively detailed biophysical population model, comprising conductance-based Hodgkin-Huxley type neuron models with connectivity rules derived from anatomical data, to various representations with fewer parameters, finishing with a firing rate network model that permits analysis. We apply this methodology to primary visual cortex of higher mammals, focusing on the functional property of stimulus orientation selectivity of receptive fields of individual neurons. The mapping produces compact expressions for the parameters of the abstract model that clearly identify the impact of specific electrophysiological and anatomical parameters on the analytical results, in particular as manifested by specific functional signatures of visual cortex, including input-output sharpening, conductance invariance, virtual rotation and the tilt after effect. Importantly, qualitative differences between model behaviours point out consequences of various simplifications. The strategy may be applied to other neuronal systems with appropriate modifications.

Optogenetic Low-Frequency Stimulation of Principal Neurons, but Not Parvalbumin-Positive Interneurons, Prevents Generation of Ictal Discharges in Rodent Entorhinal Cortex in an In Vitro 4-Aminopyridine Model.

Low-frequency electrical stimulation is used to treat some drug-resistant forms of epilepsy. Despite the effectiveness of the method in suppressing seizures, there is a considerable risk of side effects. An optogenetic approach allows the targeting of specific populations of neurons, which can increase the effectiveness and safety of low-frequency stimulation. In our study, we tested the efficacy of the suppression of ictal activity in entorhinal cortex slices in a 4-aminopyridine model with three variants of low-frequency light stimulation (LFLS): (1) activation of excitatory and inhibitory neurons (on Thy1-ChR2-YFP mice), (2) activation of inhibitory interneurons only (on PV-Cre mice after virus injection with channelrhodopsin2 gene), and (3) hyperpolarization of excitatory neurons (on Wistar rats after virus injection with archaerhodopsin gene). Only in the first variant did simultaneous LFLS of excitatory and inhibitory neurons replace ictal activity with interictal activity. We suggest that LFLS caused changes in the concentration gradients of K+ and Na+ cations across the neuron membrane, which activated Na-K pumping. According to the mathematical modeling, the increase in Na-K pump activity in neurons induced by LFLS led to an antiepileptic effect. Thus, a less specific and generalized optogenetic effect on entorhinal cortex neurons was more effective in suppressing ictal activity in the 4-aminopyridine model.

Refractory density model of cortical direction selectivity: Lagged-nonlagged, transient-sustained, and On-Off thalamic neuron-based mechanisms and intracortical amplification.

A biophysically detailed description of the mechanisms of the primary vision is still being developed. We have incorporated a simplified, filter-based description of retino-thalamic visual signal processing into the detailed, conductance-based refractory density description of the neuronal population activity of the primary visual cortex. We compared four mechanisms of the direction selectivity (DS), three of them being based on asymmetrical projections of different types of thalamic neurons to the cortex, distinguishing between (i) lagged and nonlagged, (ii) transient and sustained, and (iii) On and Off neurons. The fourth mechanism implies a lack of subcortical bias and is an epiphenomenon of intracortical interactions between orientation columns. The simulations of the cortical response to moving gratings have verified that first three mechanisms provide DS to an extent compared with experimental data and that the biophysical model realistically reproduces characteristics of the visual cortex activity, such as membrane potential, firing rate, and synaptic conductances. The proposed model reveals the difference between the mechanisms of both the intact and the silenced cortex, favoring the second mechanism. In the fourth case, DS is weaker but significant; it completely vanishes in the silenced cortex.DS in the On-Off mechanism derives from the nonlinear interactions within the orientation map. Results of simulations can help to identify a prevailing mechanism of DS in V1. This is a step towards a comprehensive biophysical modeling of the primary visual system in the frameworks of the population rate coding concept.

Insertion of Calcium-Permeable AMPA Receptors during Epileptiform Activity In Vitro Modulates Excitability of Principal Neurons in the Rat Entorhinal Cortex.

Epileptic activity leads to rapid insertion of calcium-permeable α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid receptors (CP-AMPARs) into the synapses of cortical and hippocampal glutamatergic neurons, which generally do not express them. The physiological significance of this process is not yet fully understood; however, it is usually assumed to be a pathological process that augments epileptic activity. Using whole-cell patch-clamp recordings in rat entorhinal cortex slices, we demonstrate that the timing of epileptiform discharges, induced by 4-aminopyridine and gabazine, is determined by the shunting effect of Ca2+-dependent slow conductance, mediated predominantly by K+-channels. The blockade of CP-AMPARs by IEM-1460 eliminates this extra conductance and consequently increases the rate of discharge generation. The blockade of NMDARs reduced the additional conductance to a lesser extent than the blockade of CP-AMPARs, indicating that CP-AMPARs are a more significant source of intracellular Ca2+. The study's main findings were implemented in a mathematical model, which reproduces the shunting effect of activity-dependent conductance on the generation of discharges. The obtained results suggest that the expression of CP-AMPARs in principal neurons reduces the discharge generation rate and may be considered as a protective mechanism.

Correction: Minimal model of interictal and ictal discharges "Epileptor-2".

[This corrects the article DOI: 10.1371/journal.pcbi.1006186.].

Spatial propagation of interictal discharges along the cortex.

Interictal discharges (IIDs) accompany epileptic seizures and highlight the mechanisms of pathological activity. The propagation of IIDs along the neural tissue is not well understood. To simulate IID propagation, this study proposes a new mathematical model that uses the conductance-based refractory density approach for glutamatergic and GABAergic neuronal populations. The mathematical model is found to be consistent with experimental double-patch registrations in the 4-aminopyridine in vitro model of epilepsy. In slices, the spontaneous activity of interneurons leads to their synchronization by means of the depolarizing GABAmediated response, thus initiating IIDs. Modeling reveals a clustering of interneuronal synchronization followed by IIDs with activity fronts that propagate along the cortex. The GABA-mediated depolarization either remains to be subthreshold for the principal neurons and thus results in pure GABAergic IIDs (IID1s) or leads to glutamatergic excitation, thus resulting in another type of IIDs (IID2s). In both the model and experiment, IIDs propagate as waves, with constant activity profiles and velocity. The speed of IIDs is of the order of tens of mm/s and is larger for IID2s than for IID1s (40 and 20 mm/s, respectively). The simulations, consistent with experimental observations, show that the wavelike propagation of IIDs initiated by interneurons is determined by local synaptic connectivity under the conditions of depolarizing GABA.

Minimal model of interictal and ictal discharges "Epileptor-2".

Seizures occur in a recurrent manner with intermittent states of interictal and ictal discharges (IIDs and IDs). The transitions to and from IDs are determined by a set of processes, including synaptic interaction and ionic dynamics. Although mathematical models of separate types of epileptic discharges have been developed, modeling the transitions between states remains a challenge. A simple generic mathematical model of seizure dynamics (Epileptor) has recently been proposed by Jirsa et al. (2014); however, it is formulated in terms of abstract variables. In this paper, a minimal population-type model of IIDs and IDs is proposed that is as simple to use as the Epileptor, but the suggested model attributes physical meaning to the variables. The model is expressed in ordinary differential equations for extracellular potassium and intracellular sodium concentrations, membrane potential, and short-term synaptic depression variables. A quadratic integrate-and-fire model driven by the population input current is used to reproduce spike trains in a representative neuron. In simulations, potassium accumulation governs the transition from the silent state to the state of an ID. Each ID is composed of clustered IID-like events. The sodium accumulates during discharge and activates the sodium-potassium pump, which terminates the ID by restoring the potassium gradient and thus polarizing the neuronal membranes. The whole-cell and cell-attached recordings of a 4-AP-based in vitro model of epilepsy confirmed the primary model assumptions and predictions. The mathematical analysis revealed that the IID-like events are large-amplitude stochastic oscillations, which in the case of ID generation are controlled by slow oscillations of ionic concentrations. The IDs originate in the conditions of elevated potassium concentrations in a bath solution via a saddle-node-on-invariant-circle-like bifurcation for a non-smooth dynamical system. By providing a minimal biophysical description of ionic dynamics and network interactions, the model may serve as a hierarchical base from a simple to more complex modeling of seizures.

Conductance-Based Refractory Density Approach for a Population of Bursting Neurons.

The conductance-based refractory density (CBRD) approach is a parsimonious mathematical-computational framework for modelling interacting populations of regular spiking neurons, which, however, has not been yet extended for a population of bursting neurons. The canonical CBRD method allows to describe the firing activity of a statistical ensemble of uncoupled Hodgkin-Huxley-like neurons (differentiated by noise) and has demonstrated its validity against experimental data. The present manuscript generalises the CBRD for a population of bursting neurons; however, in this pilot computational study, we consider the simplest setting in which each individual neuron is governed by a piecewise linear bursting dynamics. The resulting population model makes use of slow-fast analysis, which leads to a novel methodology that combines CBRD with the theory of multiple timescale dynamics. The main prospect is that it opens novel avenues for mathematical explorations, as well as, the derivation of more sophisticated population activity from Hodgkin-Huxley-like bursting neurons, which will allow to capture the activity of synchronised bursting activity in hyper-excitable brain states (e.g. onset of epilepsy).

Reduced Efficacy of the KCC2 Cotransporter Promotes Epileptic Oscillations in a Subiculum Network Model.

Pharmacoresistant epilepsy is a chronic neurological condition in which a basal brain hyperexcitability results in paroxysmal hypersynchronous neuronal discharges. Human temporal lobe epilepsy has been associated with dysfunction or loss of the potassium-chloride cotransporter KCC2 in a subset of pyramidal cells in the subiculum, a key structure generating epileptic activities. KCC2 regulates intraneuronal chloride and extracellular potassium levels by extruding both ions. Absence of effective KCC2 may alter the dynamics of chloride and potassium levels during repeated activation of GABAergic synapses due to interneuron activity. In turn, such GABAergic stress may itself affect Cl- regulation. Such changes in ionic homeostasis may switch GABAergic signaling from inhibitory to excitatory in affected pyramidal cells and also increase neuronal excitability. Possibly these changes contribute to periodic bursting in pyramidal cells, an essential component in the onset of ictal epileptic events. We tested this hypothesis with a computational model of a subicular network with realistic connectivity. The pyramidal cell model explicitly incorporated the cotransporter KCC2 and its effects on the internal/external chloride and potassium levels. Our network model suggested the loss of KCC2 in a critical number of pyramidal cells increased external potassium and intracellular chloride concentrations leading to seizure-like field potential oscillations. These oscillations included transient discharges leading to ictal-like field events with frequency spectra as in vitro Restoration of KCC2 function suppressed seizure activity and thus may present a useful therapeutic option. These simulations therefore suggest that reduced KCC2 cotransporter activity alone may underlie the generation of ictal discharges. SIGNIFICANCE STATEMENT: Ion regulation in the brain is a major determinant of neural excitability. Intracellular chloride in neurons, a partial determinant of the resting potential and the inhibitory reversal potentials, is regulated together with extracellular potassium via kation chloride cotransporters. During temporal lobe epilepsy, the homeostatic regulation of intracellular chloride is impaired in pyramidal cells, yet how this dysregulation may lead to seizures has not been explored. Using a realistic neural network model describing ion mechanisms, we show that chloride homeostasis pathology provokes seizure activity analogous to recordings from epileptogenic brain tissue. We show that there is a critical percentage of pathological cells required for seizure initiation. Our model predicts that restoration of the chloride homeostasis in pyramidal cells could be a viable antiepileptic strategy.

Conductance-based refractory density approach: comparison with experimental data and generalization to lognormal distribution of input current.

The conductance-based refractory density (CBRD) approach is an efficient tool for modeling interacting neuronal populations. The model describes the firing activity of a statistical ensemble of uncoupled Hodgkin-Huxley-like neurons, each receiving individual Gaussian noise and a common time-varying deterministic input. However, the approach requires experimental validation and extension to cases of distributed input signals (or input weights) among different neurons of such an ensemble. Here the CBRD model is verified by comparing with experimental data and then generalized for a lognormal (LN) distribution of the input weights. The model with equal weights is shown to reproduce efficiently the post-spike time histograms and the membrane voltage of experimental multiple trial response of single neurons to a step-wise current injection. The responses reveal a more rapid reaction of the firing-rate than voltage. Slow adaptive potassium channels strongly affected the shape of the responses. Next, a computationally efficient CBRD model is derived for a population with the LN input weight distribution and is compared with the original model with equal input weights. The analysis shows that the LN distribution: (1) provides a faster response, (2) eliminates oscillations, (3) leads to higher sensitivity to weak stimuli, and (4) increases the coefficient of variation of interspike intervals. In addition, a simplified firing-rate type model is tested, showing improved precision in the case of a LN distribution of weights. The CBRD approach is recommended for complex, biophysically detailed simulations of interacting neuronal populations, while the modified firing-rate type model is recommended for computationally reduced simulations.

A mathematical model of color and orientation processing in V1.

Orientation processing in the primary visual cortex (V1) has been experimentally investigated in detail and reproduced in models, while color processing remains unclear. Thus, we have constructed a mathematical model of color and orientation processing in V1. The model is mainly based on the following experimental evidence concerning color blobs: A blob contains overlapping neuronal patches activated by different hues, so that each blob represents a full gamut of hue and might be structured with a loop (Xiao et al. in NeuroImage 35:771-786, 2007). The proposed model describes a set of orientation hypercolumns and color blobs, in which color and orientation preferences are represented by the poloidal and toroidal angles of a torus, correspondingly. The model consists of color-insensitive (CI) and color-sensitive (CS) neuronal populations, which are described by a firing-rate model. The set of CI neurons is described by the classical ring model (Ben-Yishai et al. in Proc Natl Acad Sci USA 92:3844-3848, 1995) with recurrent connections in the orientation space; similarly, the set of CS neurons is described in the color space and also receives input from CI neurons of the same orientation preference. The model predictions are as follows: (1) responses to oriented color stimuli are significantly stronger than those to non-oriented color stimuli; (2) the activity of CS neurons in total is higher than that of CI neurons; (3) a random color can be illusorily perceived in the case of gray oriented stimulus; (4) in response to two-color stimulus in the marginal phase, the network chooses either one of the colors or the intermediate color; (5) input to a blob has rather continual representation of a hue than discrete one (with two narrowly tuned opponent signals).

Conductance-based refractory density model of primary visual cortex.

A layered continual population model of primary visual cortex has been constructed, which reproduces a set of experimental data, including postsynaptic responses of single neurons on extracellular electric stimulation and spatially distributed activity patterns in response to visual stimulation. In the model, synaptically interacting excitatory and inhibitory neuronal populations are described by a conductance-based refractory density approach. Populations of two-compartment excitatory and inhibitory neurons in cortical layers 2/3 and 4 are distributed in the 2-d cortical space and connected by AMPA, NMDA and GABA type synapses. The external connections are pinwheel-like, according to the orientation of a stimulus. Intracortical connections are isotropic local and patchy between neurons with similar orientations. The model proposes better temporal resolution and more detailed elaboration than conventional mean-field models. In comparison to large network simulations, it excludes a posteriori statistical data manipulation and provides better computational efficiency and minimal parametrization.

Light-Driven Sodium Pump as a Potential Tool for the Control of Seizures in Epilepsy.

The marine flavobacterium Krokinobactereikastus light-driven sodium pump (KR2) generates an outward sodium ion current under 530 nm light stimulation, representing a promising optogenetic tool for seizure control. However, the specifics of KR2 application to suppress epileptic activity have not yet been addressed. In the present study, we investigated the possibility of KR2 photostimulation to suppress epileptiform activity in mouse brain slices using the 4-aminopyrindine (4-AP) model. We injected the adeno-associated viral vector (AAV-PHP.eB-hSyn-KR2-YFP) containing the KR2 sodium pump gene enhanced with appropriate trafficking tags. KR2 expression was observed in the lateral entorhinal cortex and CA1 hippocampus. Using whole-cell patch clamp in mouse brain slices, we show that KR2, when stimulated with LED light, induces a substantial hyperpolarization of entorhinal neurons. However, continuous photostimulation of KR2 does not interrupt ictal discharges in mouse entorhinal cortex slices induced by a solution containing 4-AP. KR2-induced hyperpolarization strongly activates neuronal HCN channels. Consequently, turning off photostimulation resulted in HCN channel-mediated rebound depolarization accompanied by a transient increase in spontaneous network activity. Using low-frequency pulsed photostimulation, we induced the generation of short HCN channel-mediated discharges that occurred in response to the light stimulus being turned off; these discharges reliably interrupt ictal activity. Thus, low-frequency pulsed photostimulation of KR2 can be considered as a potential tool for controlling epileptic seizures.

Response retention and apparent motion effect in visual cortex models.

Apparent motion is a visual illusion in which stationary stimuli, flashing in distinct spatial locations at certain time intervals, are perceived as one stimulus moving between these locations. In the primary visual cortex, apparent-motion stimuli produce smooth spatio-temporal patterns of activity similar to those produced by continuously moving stimuli. An important prerequisite for producing such activity patterns is prolongation of responses to brief stimuli. Indeed, a brief stimulus can evoke in the visual cortex a long response, outlasting the stimulus by hundreds of milliseconds. Here we use firing-rate based models with simple ring structure, and biologically-detailed conductance-based refractory density (CBRD) model with retinotopic space representation to analyze the response retention and the origin of smooth profiles of activity in response to apparent-motion stimuli. We show that the strength of recurrent connectivity is the major factor that endorses neuronal networks with the ability for response retention. The same strengths of recurrent connections mediate the appearance of bump attractor in the ring models. Factors such as synaptic depression, NMDA receptor mediated currents, and conductances regulating spike adaptation influence response retention, but cannot substitute for the weakness of recurrent connections to reproduce response retention in models with weak connectivity. However, the weakness of lateral recurrent connections can be compensated by layering: in multi-layer models even with weaker connections the activity retains due to its feedforward propagation from layer to layer. Using CBRD model with retinotopic space representation we further show that smooth spatio-temporal profiles of activity in response to apparent-motion stimuli are produced in the models expressing response retention, but not in the models that fail to produce response retention. Together, these results demonstrate a link between response retention and the ability of neuronal networks to generate spatio-temporal patterns of activity, which are compatible with perception of apparent motion.

Molecular mechanisms of action determine inhibition of paroxysmal depolarizing shifts by NMDA receptor antagonists in rat cortical neurons.

N-methyl-d-aspartate glutamate receptors (NMDARs) are involved in numerous central nervous system (CNS) processes, including epileptiform activity. We used a picrotoxin-induced epileptiform activity model to compare the action of different types of NMDAR antagonists in rat brain slices. Paroxysmal depolarizing shifts (PDS) were evoked by external stimulation in the medial prefrontal cortex (mPFC) slices and recorded in pyramidal cells (PC) and in fast-spiking interneurons (FSI). The NMDAR antagonists APV and memantine reduced the duration of PDS. However, the competitive antagonist APV caused similar effects on the PC and FSI, while the open-channel blocker memantine had a much stronger effect on the PDS in the FSI than in the PC. This difference cannot be explained by a corresponding difference in NMDAR sensitivity to memantine because the drug inhibited the excitatory postsynaptic current (EPSC) similarly in both cell types. Importantly, the PDS were significantly longer in the FSI than in the PC. The degree of PDS inhibition by memantine correlated with individual PDS durations in each cell type. Computer modeling of a synaptic network in the mPFC suggests that the different effects of memantine on the PDS in the PC and FSI can be explained by use dependence of its action. An open-channel blocking mechanism and competition with Mg2+ ions for the binding site result in pronounced inhibition of the long PDS, whereas the short PDS are weakly sensitive. Our results show that peculiarities of kinetics and the mechanism of action largely determine the effects of NMDAR antagonists on physiological and/or pathological processes.

Mappings between a macroscopic neural-mass model and a reduced conductance-based model.

We present two alternative mappings between macroscopic neuronal models and a reduction of a conductance-based model. These provide possible explanations of the relationship between parameters of these two different approaches to modelling neuronal activity. Obtaining a physical interpretation of neural-mass models is of fundamental importance as they could provide direct and accessible tools for use in diagnosing neurological conditions. Detailed consideration of the assumptions required for the validity of each mapping elucidates strengths and weaknesses of each macroscopic model and suggests improvements for future development.