1.1 Proposed dorsal horn subcircuits mediating allodynia

Here, we describe subcircuit motifs that reflect recent experimental evidence for the structure of dorsal horn laminae I-II networks mediating static allodynia (as evoked by a von Frey device) and dynamic allodynia (Fig 1A). Experimental studies in rodents have shown that static allodynia is reliant on activity in three putatively different types of laminae I-II excitatory interneurons: somatostatin-positive (SOM+) [16], calretinin-positive (CR+) [19], and protein kinase C γ-positive (PKCγ+) [17] cells. Additionally, inactivation of either dynorphin-positive (DYN+) [16] or parvalbumin-positive (PV+) [17] inhibitory interneurons is sufficient to produce static allodynia, suggesting that these inhibitory cells usually gate activation of SOM+, CR+ and PKCγ+ excitatory cells. Experiments have also identified direct synaptic connections from PV+ to PKCγ+ neurons [17].

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Fig 1. Proposed subcircuits in laminae I-II of the dorsal horn mediating static and dynamic allodynia.

(A) Schematic of neural circuitry in laminae I-II of the dorsal horn consisting of populations of inhibitory (magenta shades) and excitatory (blue shades) interneurons that filter Aβ and C fiber inputs to projection neurons (green) that transmit signals to the brain. (B) A schematic of the proposed subcircuit mediating static allodynia. I1 and I2 represent the populations of dynorphin-positive (DYN+) and parvalbumin-positive (PV+) inhibitory interneurons, respectively, and E represents a collective population of somatostatin-positive (SOM+), calretinin-positive (CR+) and protein kinase C γ-positive (PKCγ+) excitatory interneurons. (C) A schematic of the proposed subcircuit mediating dynamic allodynia. I1 and I2 represent populations of DYN+ inhibitory neurons, and E1 and E2 represent the populations of VGLUT3-positive (VT3) and somatostatin-positive/calretinin-negative (SOM+/CR-) excitatory interneurons. In all panels, Aβ and C represent inputs relayed from the periphery along Aβ and C fibers, respectively.

https://doi.org/10.1371/journal.pcbi.1012234.g001

Based on these experimental results, we consider a simplified subcircuit motif representing part of the pathway mediating static allodynia that consists of three neural populations (Fig 1B). One population, E, represents the collective activity of laminae I-II SOM+, CR+ and PKCγ+ excitatory interneurons. The E population is inhibited by two distinct inhibitory interneuron populations, I1 representing DYN+ cells and I2 representing PV+ cells. All three populations receive input from Aβ fibers, (as suggested in [16, 19, 21]), while the E population is presumed to be additionally targeted by C fiber input (see e.g. [16, 19]).

Rodent experimental studies probing the mechanisms for dynamic allodynia have identified that it relies on the activity of laminae I-II SOM+ excitatory interneurons that do not express the calbindin 2/calretinin gene (CR-). Specifically, ablating SOM+ neurons abolishes or greatly reduces dynamic allodynia in mice [18]. On the other hand, ablating neurons expressing the calbindin 2/calretinin gene (CR+) had no effect on dynamic allodynia [18]. Thus, the SOM+ neurons that are necessary for dynamic allodynia must also be negative for calretinin (SOM+/CR-). Excitatory interneurons expressing vesicular glutamate transporter 3 (VT3+) in laminae II-III also contribute to dynamic allodynia since their ablation or silencing eliminates or attenuates dynamic allodynia induced by nerve injury or ablation in mice [18]. Moreover, it has been proposed that the VT3+ neurons synapse onto the excitatory SOM+/CR- neurons [18]. Further experimental results suggest that activity of the VT3+ excitatory interneurons are gated by DYN+ inhibitory neurons since VT3+ neurons fire action potentials in response to Aβ input when inhibitory signaling is blocked [18]. Ablation of DYN+ cells is sufficient to induce dynamic allodynia [16] suggesting that DYN+ inhibitory cells may provide local inhibitory control of these two excitatory interneuron populations involved in dynamic allodynia.

To account for these experimental results, we consider a simplified subcircuit representing part of the pathway mediating dynamic allodynia that consists of four neural populations (Fig 1C). An excitatory population E2 representing the collective activity of laminae I-II SOM+/CR- excitatory neurons is excited by a population E1 representing in laminae II-III VT3+ neurons. E1 and E2 receive inhibitory input from populations I1 and I2, respectively, representing local DYN+ cells. All populations receive Aβ input, (as suggested in [16, 18]), and E2 is presumed to be additionally targeted by C fiber input (see [18]).

The E population in the static allodynia subcircuit and the E2 population in the dynamic allodynia subcircuit are assumed to provide direct synaptic input to lamina I projection cells that transmit painful signals to the brain [22]. Hence, painful stimuli generating activity on C fibers would activate these excitatory cells and the downstream projection neurons. Under normal healthy conditions, responses of these excitatory populations to non-painful stimuli signaled by Aβ input is assumed to be gated by the inhibitory populations. Allodynia occurs when the E-I balance is disrupted such that non-painful Aβ input causes firing in these excitatory populations, leading to activation of the downstream projection neurons. While allodynia can also be induced by nociceptor-mediated sensitization of projection neurons [20], here we focus on allodynia caused by disruption of inhibitory gating of Aβ input.

1.2 Overview

The goal of this study is to identify and characterize “most likely” potential mechanisms that disrupt the processing of Aβ signals and E-I balance in these subcircuits leading to allodynia. We model the activity and interactions of the neural populations in these subcircuits using a population firing rate model formalism (see e.g. [23–25]) constrained by experimental data. We begin by identifying the full space of model parameters that generate responses to Aβ signals that is correlated to healthy conditions in each subcircuit, i.e. low firing of the E or E2 populations. We specifically focus on analyzing effects of synaptic coupling parameters that represent the magnitude of effects of pre-synaptic activity on post-synaptic response. Due to the relative mathematical simplicity of our firing rate model formalism, we are able to analytically define an “allowable parameter space” (APS) as a solution to a system of inequalities on model parameters. The APS for each subcircuit indicates the considerable range of parameter combinations that can reproduce healthy responses and demonstrates the myriad ways that E-I balance might be obtained. The population of subcircuits corresponding to the points in the APS represents the high variability in subcircuit structure that may occur physiologically.

Next, for each subcircuit, we define the analytic condition on model parameters that generates a response to Aβ input correlated with allodynia, i.e. high firing of the E or E2 populations. Solving this condition defines a hypersurface in the model parameter space, which we call the “allodynia surface”, that separates parameter sets for which the subcircuit generates healthy, allodynia-free responses for all typical Aβ inputs, vs responses where allodynia may be present, (for at least some typical level of Aβ input). For model parameter sets (points) in the APS, we then identify the minimal change in parameters that leads to allodynia by computing the shortest vector in parameter space from the point in the APS to the allodynia surface. The direction of these shortest vectors, corresponding to changes in specific model parameters, indicates the least variation of the subcircuit structure that results in the allodynia response. Since larger variations in parameter values can also generate an allodynia response, these shortest vectors represent the “most likely” potential mechanisms for allodynia to occur.

We then analyze the directions of the shortest vectors from the APS to the allodynia surface to identify distinct variations of parameters that dysregulate E-I balance leading to allodynia. Specifically, we find that the shortest vector directions form clusters based on the parameters requiring the relatively smallest changes to reach the allodynia surface. These different clusters can be interpreted as representing different underlying mechanisms for generating allodynia.

We find that these proposed allodynia mechanisms generally involve the dysregulation of E-I balance occurring due to the release of excitatory cells from inhibitory gating or to excitatory cells escaping inhibitory gating. We interpret “release” as reflecting a decrease in inhibitory population firing and “escape” as including changes to excitatory population responses to inhibitory population firing, as well as increased excitatory population firing. Significantly, the specific subcircuit components that are associated with “release” or “escape” mechanisms differ in the subcircuits due to their differing network structures. As such, our results identify the diverse ways by which excitatory and inhibitory components within the subcircuits combine and interact to maintain E-I balance, and characterize the sensitivity of these subcircuits to disruptions that can lead to pathological responses.

The paper is organized as follows: in Section 2.1, in order to illustrate our analysis method, we first analyze a simple subcircuit representing the canonical gate control model. In Sections 2.2 and 2.3, we then apply the analysis to the proposed subcircuits mediating static and dynamic allodynia, respectively. In Section 3, we summarize the results and put them into a greater physiological context. Details of the models and analysis methods are contained in Section 4.

2.1 Analysis of a simple “gate control” subcircuit

We first illustrate our analysis methodology by applying it to a simple subcircuit representing the canonical “gate control” model (Fig 2A). This simple subcircuit consists of an inhibitory interneuron population (I) that inhibits an excitatory interneuron population (E). Both populations receive input from Aβ fibers. We assume that any sustained activity of the E population is a proxy for a painful response. Thus, under healthy conditions, Aβ activity in the range of 10–20 Hz [27] should not cause sustained firing of the E population. This is achieved through appropriate balancing of the inhibitory input from the I population and the excitatory Aβ input onto the E population. We characterize different ways in which this subcircuit can become dysregulated to produce firing in the E population in response to normal Aβ input.

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Fig 2. Allowable parameter space (APS) for the simple “gate control” subcircuit.

(A) A schematic of the simple “gate control” subcircuit, where E and I represent populations of excitatory and inhibitory interneurons, respectively. (B) Simulation results showing the mean (black lines) and range (shaded gray areas) for the firing-rate responses of the I population (left panel), and the average voltage (middle panel) and firing rate (right panel) of the E population, simulated for 20 sampled points in the APS each with a different random Aβ stimulus in range [10, 20] Hz (during t ∈ [0.2, 0.7] s). (C) Violin plots showing the distribution of each coupling strength parameter with mean (square marker) and range of values that lie within one standard deviation (vertical bar) indicated. (D) A scatter plot of 5000 uniformly sampled APS points in the normalized space. (E) Normalized Pearson correlation coefficients between coupling strength parameters in the normalized APS sample.

https://doi.org/10.1371/journal.pcbi.1012234.g002

2.1.1 The APS for the simple subcircuit.

We begin by deriving conditions on the coupling strength parameters in the subcircuit, namely (g_AβI, g_IE, g_AβE), that generate normal healthy responses to Aβ inputs to define the APS. Specifically, we impose conditions on the steady-state voltages of the E and I populations in response to constant Aβ input in the range of 10–20 Hz to account for experimental observations. To start, we require that steady-state voltages of the E and I populations, and , respectively, remain within biologically reasonable bounds, V_min and V_max, in response to a sustained, typical non-painful input on the Aβ fibers (v_I and v_E upper bounds in Table 1). To incorporate the phenomenon of pain inhibition, or Aβ input gating responses to C fiber input, we require that the net signaling to the E population in response to typical Aβ activity is inhibitory. To enforce this gating response under normal healthy conditions, we require that the I population fires for all typical, non-painful f_Aβ values, but the E population does not, thus representing a non-painful response (pain inhibition condition in Table 1). Similarly, we enforce that the E population does not fire for low but non-zero f_Aβ values (which we take to be in [0, 10] Hz). Finally, we incorporate experimental results showing that ablation of DYN+ inhibitory interneurons causes allodynia [16] by requiring that the E population fires in response to typical Aβ input in the absence of I population activity (I ablation condition in Table 1).

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Table 1. Conditions and resulting inequalities used to constrain the simple subcircuit and define its allowable parameter space (APS).

Condition type and Condition (first 2 columns) describe the rationale behind each condition. The resulting inequality on population steady-state voltages and are given in the 3rd column. These inequalities must hold for typical non-painful f_Aβ values, which we take to be f_Aβ ∈ [10, 20] Hz. V_x,max and V_x,min are the maximum and minimum limits on average voltage, respectively, V_x,rest is the resting membrane voltage and V_x,thr is the voltage threshold for firing (x = E, I).

https://doi.org/10.1371/journal.pcbi.1012234.t001

These conditions can be re-written as a system of inequalities, some nonlinear, on the coupling strengths (g_AβI, g_IE and g_AβE) (see Section 4.4.1), the solution of which constitutes the APS for this subcircuit. Because these inequalities must be satisfied for a range of f_Aβ input values, solving them explicitly requires finding the solution to a system of nonlinear optimization problems. As described in Section 4.4.1, this system for the simple subcircuit can be explicitly solved using Lambert functions. To obtain the APS, instead of using this analytic solution, we compute a uniformly-distributed sample using a custom sampling algorithm (see Section 4.6).

To illustrate that the points in the APS correspond to subcircuit instantiations that yield the desired behaviors, we simulate instances of the simple subcircuit model for 20 randomly sampled APS points in response to a noisy Aβ signal with mean firing rate in [10, 20] Hz. The subcircuit responses show I population firing that inhibits the E population such that it does not exhibit sustained firing (Fig 2B). Fig 2C shows the distributions of coupling strength values across the APS. Because g_AβI can be changed the most without exiting the APS, whereas g_IE can be changed the least, they are in some sense the least and most sensitive coupling strengths, respectively, maintaining E-I balance in this subcircuit. To fairly compare relative changes in g_yx (y, x = E, I, Aβ), we consider normalized coupling strengths as a proportion of their observed ranges over the APS, , and continue our analysis using the normalized APS in (, , ) space, as shown in Fig 2D.

To identify mechanisms that contribute to the maintenance and eventual disruption of E-I balance, we computed the correlations among normalized coupling strength values in the APS. Fig 2E shows that and values are strongly positively correlated, highlighting that direct excitatory and inhibitory inputs to the E population are balanced to prevent firing in the E population. Additionally, and are negatively correlated, demonstrating the multi-synaptic regulation of feedforward inhibitory signaling to the E population. The slight negative correlation between and also suggests a balance in the subcircuit of excitatory and inhibitory responses to Aβ signaling.

2.1.2 Mechanisms for generating allodynia in the simple subcircuit.

To determine the vulnerabilities of the simple subcircuit to allodynia, we find which changes in the coupling strengths most easily result in sustained E population firing in response to non-painful f_Aβ input (in the range [10, 20] Hz). In our model formalism, allodynia is presumed to occur when the steady-state average voltage of the E population, , exceeds the firing threshold for some typical non-painful, sustained f_Aβ:

We use this inequality to define the allodynia surface S in (g_AβI, g_IE, g_AβE) space, above which the corresponding subcircuit instantiation produces allodynia for at least one value of f_Aβ ∈ [10, 20] Hz. We use “above” in the sense that the g_AβE component of a point in parameter space defines a height for that point. This surface, S, is then the following set of points:

We compute S by solving this minimization problem in the unnormalized parameter space (see Section 4.7 for details). Plotting S in the normalized space shows that it always lies above the APS (Fig 3A).

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Fig 3. Allodynia mechanisms represented as shortest paths from the APS to the allodynia surface in the simple “gate control” subcircuit.

(A) A scatter plot of the sampled APS points (5000 points) in the normalized space with the allodynia surface S overlaid. Based on the direction of their shortest paths, APS points separate into 2 clusters (Cluster 1 (red) and Cluster 2 (blue)). The nearest point on the allodynia surface from each sampled APS point is also shown (darker red and blue points on S). (B) Violin plots of the coupling strength distributions for the APS points in each cluster. Black * shows the mean APS values, colored square shows the mean values in each cluster. (C) The probability distribution of the shortest distances to the allodynia surface (overall profile). The shading represents the contributions from each cluster to the overall profile. For instance, a bar that is 70% light blue indicates that cluster 2 constitutes 70% of subcircuit instantiations with the corresponding distance to the allodynia surface. (D) Parallel plot representation of the components of the shortest path vectors from APS points to their corresponding nearest points on S, colored according to cluster membership. (E) Firing rate responses f_E (left panels) and f_I (right panels) to a noisy Aβ input signal with amplitude f_Aβ ∈ [10, 20] Hz occurring during t ∈ [0.2, 0.7]s for each cluster, (top row: Cluster 1, bottom row: Cluster 2). Solid lines correspond to the subcircuit with coupling strength values set to the mean values for the cluster and dash-dotted lines correspond to the simple subcircuit instantiation with coupling strengths set to the corresponding closest point on the allodynia surface. Aβ input mean frequencies are chosen as the smallest value that induces allodynia for each cluster.

https://doi.org/10.1371/journal.pcbi.1012234.g003

For each subcircuit instantiation associated with a point in the APS, we identify its vulnerability to allodynia by computing the shortest path from that point to the allodynia surface S using a customized global optimization scheme (see Section 4.8). Because the shortest path indicates how to reach the allodynia surface by altering coupling strengths as little as possible, it represents the direction in parameter space in which the subcircuit instantiation is most vulnerable to allodynia. In this way, the components of the shortest path vector suggest which coupling strengths will need to change, along with the corresponding magnitudes of their relative change, in order to disrupt E-I balance and induce allodynia.

Clustering of the shortest path vectors, using a density-based scanning algorithm [28], identifies two clusters in the APS for the simple subcircuit (Fig 3A): Cluster 1 (red) contains APS points for which primarily decreasing will reach the allodynia surface in minimal distance, and Cluster 2 (cyan) contains points whose shortest path involves decreasing and increasing . Note that some points in the APS, especially along the border of these clusters, may reach the allodynia surface in different directions but with similar length vectors. To ensure our clusters are robust, we additionally computed the paths corresponding to nearby local minima in the distances from APS points to the allodynia surface, only considering local minima within 1.5 times the global minimum distance to avoid considering changes unlikely to happen (such as by increasing g_AβE by an unreasonable amount like 300%). With this expanded set of paths, the same two clusters were obtained (S1 Fig, panel A), demonstrating that while individual subcircuits in the APS may reach the allodynia surface through different directions, all minimal distance directions fall into these two clusters.

APS points in Cluster 1 are characterized by smaller values of and larger values of and on average, relative to the whole APS, while points in Cluster 2 have the opposite relative values (Fig 3B). We also see that points in Cluster 1 are generally closer to the allodynia surface than Cluster 2 (Fig 3C), indicating that points in Cluster 1 are more sensitive to dysregulation than points in Cluster 2. In terms of the most likely mechanism for allodynia, the shortest paths from points in Cluster 1 to S consist of decreasing and increasing by a relatively smaller amount (Fig 3D), with larger decreases in g_AβI being coupled with larger increases in g_AβE (see the dark red lines in Fig 3D). For Cluster 2, the shortest paths to the allodynia surface involve decreasing and increasing (Fig 3D) in a fixed ratio (S2 Fig). Thus, the shortest paths to the allodynia surface from points in Cluster 2 are always in exactly the same direction, namely a ∼30% decrease in and a ∼20% increase in .

In terms of E-I balance, the most efficient means of producing allodynia for instances of the subcircuit in Cluster 1 involve disinhibition of the E population by lowering the response of the I population to Aβ input (by reducing ), and over-exciting the E population (by increasing )(Fig 3E). We characterize this mechanism of disrupting the E-I balance as the E population primarily being “released” from inhibitory control with a weaker “escape” effect. To illustrate this in the model, we show the temporal firing rates of the E and I populations in response to a noisy Aβ input (top panels in Fig 3E). Notice that a typical subcircuit instantiation in Cluster 1 with parameter values drawn from the APS (solid curves) exhibits I population firing rates around 50 Hz during non-painful Aβ input and no E population firing. However, when coupling strengths are set to the associated closest point on S (dash-dotted curves), I population firing rates are much lower and the E population is “released” from inhibitory control and is able to fire.

In contrast, the most efficient way to induce allodynia for points in Cluster 2 involves a lowering of the effect of I population activity on the E population through a reduction of coupled with an over-excitation of the E population through an increase in . We characterize this mechanism for allodynia as the E population “escaping” inhibitory control. In particular, we interpret decreased as representing a post-synaptic mechanism within the E population that decreases the effect of the I population input, hence an “escape” of the E population from inhibitory control. An example firing rate response shows that for a typical subcircuit instantiation in Cluster 2 (bottom panels in Fig 3E), the I population firing rate is saturated at its maximum value (80 Hz) during Aβ input for coupling strengths in the APS and at their closest values on the allodynia surface, thus I population activity is not altered in the allodynia condition. However, in the allodynia case, the E population fires due to an “escape” from this inhibitory control.

For the simple subcircuit, the APS is basically evenly split into the 2 clusters with ∼52% of points in Cluster 1 and ∼48% in Cluster 2, suggesting that the release and escape mechanisms for allodynia are equally likely to occur. While these mechanisms for allodynia are intuitively clear and perhaps unsurprising, as shown below for the static and dynamic allodynia subcircuits, the most likely allodynia mechanism may be biased towards one of these mechanisms and can be generated by different subcircuit components when the subcircuit structure is more complex.

2.2 Analysis of the subcircuit mediating static allodynia

The static subcircuit consists of two inhibitory populations (I1 and I2), and one excitatory (E) population, all receiving Aβ input (Fig 4A). As in the simple circuit, we assume that the E population directly relays signals to projection neurons and allodynia is defined as any sustained firing of the E population in response to typical non-painful Aβ input. The APS is defined in the 5-dimensional space of coupling parameters (g_AβI1, g_I1E, g_AβE, g_AβI2, g_I2E) and the allodynia surface is a hypersurface in this space.

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Fig 4. Allowable parameter space (APS) for the proposed subcircuit mediating static allodynia.

(A) A schematic of the static subcircuit. I1 and I2 represent the populations of inhibitory neurons and E represents the population of excitatory neurons that synapse onto projection neurons. Aβ represents inputs to the subcircuit relayed from the periphery along Aβ fibers. (B) The mean (black lines) and range (shaded gray areas) for the firing rate (top left) and voltage (top right) of the E population, as well as for the firing rates of the I1 (bottom left) and I2 (bottom right) populations calculated from 20 sampled sets of coupling strengths each with a different random Aβ input stimulus (active during t ∈ [0.2, 0.7] s). (C) Violin plots showing the distribution of each coupling strength parameter with mean (square marker) and range of values that lie within one standard deviation (vertical bar) indicated. (D) Parallel plot representation of the sampled sets of normalized coupling strengths. A line gives the values of each coupling strength in a sampled set, colored on a gradient from light red to dark red according to g_AβI1 so as to easier differentiate between individual lines.

https://doi.org/10.1371/journal.pcbi.1012234.g004

While the I1 and I2 populations represent classes of inhibitory interneurons with different molecular markers (DYN+ and PV+, respectively), we assume they have similar response properties [26] and use the same model parameters for each population. In this way, the subcircuit is symmetric in the sense that there is nothing to distinguish the two inhibitory populations from one another. As shown below, this symmetry is reflected in the structure of the APS for this subcircuit, as well as in features of the predicted most likely mechanisms for allodynia. However, in contrast to the simple subcircuit, the most likely allodynia mechanisms are biased towards different modes of E population release and escape from inhibitory control.

2.2.1 The APS for the static subcircuit.

To define the APS, we impose conditions on the steady-state voltages of the E, I1, and I2 populations so that all instantiations of the static subcircuit display desired experimentally identified behaviors. In contrast to the simple circuit, since ablation of either inhibitory population can induce allodynia, we require both inhibitory populations to be active to maintain pain inhibition. Thus, under control conditions, typical Aβ stimuli induce both I1 and I2 firing, preventing E from firing, and reducing V_E below its resting voltage. However, if one I population is ablated, then the excitatory signaling from Aβ input is strong enough to overcome the inhibition from the remaining inhibitory population to induce E firing.

These conditions are written as inequalities on steady-state population voltages, summarized in Table 2, and then re-written as a set of inequalities for the coupling strengths that must be satisfied for all f_Aβ input levels in the range [10, 20] Hz (see Section 4.4.2). The APS for the static subcircuit is then defined as the sets of 5-tuples of coupling strengths (g_AβI1, g_I1E, g_AβE, g_AβI2, g_I2E) which satisfy the system of inequalities and optimization problems. As discussed in Section 4.4.2, we simplify the optimization problems using Lambert functions and then uniformly sample from the defined APS using our customized sampling algorithm (Section 4.6).

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Table 2. Conditions that the proposed subcircuit mediating static allodynia must satisfy and the resulting inequalities on steady-state voltages.

We ensure that the subcircuit exhibits these behaviors by imposing conditions (middle column) on the subcircuit exhibited in either control, I1-ablation, or I2-ablation conditions (left-most column), and is realized as an inequality (right-most column) on the steady-state voltage of a population. Inequalities must be satisfied for f_Aβ ∈ [10, 20] Hz, unless otherwise noted.

https://doi.org/10.1371/journal.pcbi.1012234.t002

Firing rates in subcircuits for 20 randomly sampled APS points in response to a noisy Aβ signal with mean firing rate in [10, 20] Hz show that inhibition from I1 and I2 prevent sustained E population firing and cause E voltage to drop, as required for pain inhibition (Fig 4B). There is, however, initial transient E population firing due to high g_AβE values. This initial transient firing can be reduced if the time constant ratio of the E and I populations is increased. Violin plots (Fig 4C) of the distributions of coupling strengths in the APS highlight the symmetry of the static subcircuit. In particular, the distributions of g_AβI1 and g_AβI2 are similar, as are those for g_I1E and g_I2E. The values of g_AβE are large in order to balance out the two sources of inhibition in this subcircuit. The small range of g_AβE values reflects tight constraints on population responses by the conditions in Table 2 when a second inhibitory population participates in control of the E population.

The APS for the static subcircuit demonstrates several pathways for maintaining and eventually destroying E-I balance. As suggested by the biomodality in the violin plots (Fig 4C), some APS points have larger values of g_I1E but smaller values of g_I2E (dark red lines in Fig 4D with low g_AβI1) while other points show the reverse (lighter red lines in Fig 4D with higher g_AβI1). Values of g_AβI1 and g_AβI2 are negatively correlated to g_I1E and g_I2E, respectively, in these APS groups reflecting a balance to maintain sufficient inhibitory control of the E population.

2.2.2 Mechanisms of allodynia in the static subcircuit.

We consider the allodynia surface for the static subcircuit S_stat (formally defined in Section 4.7) as the set of points in normalized parameter space above which subcircuit instantiations produce allodynia for at least some non-painful f_Aβ value. Computing the shortest vectors from each sampled point in the APS to S_stat and clustering based on their directions (see Section 4.9) results in two clusters (Fig 5A). The APS points in Cluster 1 (green) are characterized by larger and smaller , and the opposite relation between and values (top panel) while the APS points in Cluster 2 (blue) are characterized by larger and smaller , and the opposite relation between and values (bottom panel), compared to mean APS values. Notably, Clusters 1 and 2 correspond to the groups of APS points that have large and large values, respectively (compare Figs 5A to 4D), and both clusters are represented almost equally in the APS (Fig 5B).

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Fig 5. Mechanisms for static allodynia represented as shortest paths to the allodynia surface for the static allodynia subcircuit.

(A) Violin plots of coupling strength distributions for the APS points in the 2 clusters of directions of shortest paths for the static subcircuit: Cluster 1 (green, top panel), Cluster 2 (blue, bottom panel). Black * represents the mean APS coupling strength values, colored square is mean value for the cluster. (B) Probability distribution of the shortest distances to the allodynia surface (overall profile); shading represents the contributions from each cluster to the overall distribution. For instance, a bar that is 60% blue indicates that cluster 2 constitutes 60% of subcircuit instantiations with the corresponding distance to the allodynia surface. (C) Parallel plot representation of the directions of the shortest paths to the allodynia surface from each sampled point in the APS. A line indicates the components of the displacement vector corresponding to one shortest path. Lines are colored based on clusters with a color gradient for better visualization. (D) Firing-rate responses to a noisy Aβ input signal (active during t ∈ [0.2, 0.7] s) for each population (columns) and for each cluster (rows). Solid lines correspond to a subcircuit instantiation with coupling strengths given by the mean values for the particular cluster, and dash-dotted lines correspond to the subcircuit instantiations with coupling strengths given by the corresponding closest point on the allodynia surface. Aβ input frequencies are chosen as the smallest value that induces allodynia for each cluster.

https://doi.org/10.1371/journal.pcbi.1012234.g005

In terms of mechanisms for allodynia, both clusters reflect the release from one source of inhibitory control with a weaker escape effect from the other source of inhibition (Fig 5C). Cluster 1 subcircuits display high I1 population activity whose inhibitory signaling is tempered by low values, and a weakly active I2 population that relies on a high value for a sufficient inhibitory effect on the E population (Fig 5D, top panel). The most efficient way to induce allodynia in Cluster 1 subcircuits is to decrease values, releasing the E population from the inhibitory control provided by the weakly active I2 population. The shortest vectors in Cluster 1 also include a slight decrease in reflecting a small simultaneous escape by E from the inhibitory control of the more strongly active I1 population. On the other hand, in Cluster 2 subcircuit instantiations, the I1 population shows lower firing rates than I2 (Fig 5D, bottom panel) and the most efficient way to produce allodynia is through decreasing values, thus releasing E from the weaker I1 inhibitory control. Shortest path vectors also include a small, simultaneous escape by E from inhibitory control of the strongly active I2 population. This mirror symmetry in the allodynia mechanisms of the two clusters is clearly reflected in the components of their shortest path vectors (Fig 5C).

In the static circuit, a solely escape mechanism involving an increase in values, combined with decreases in either or , is not the most efficient way to induce allodynia. We note that there are directions of paths from some APS points in each cluster that are within 1.5 times the distance of the shortest path that additionally include increases in (S1 Fig, panel B). However, these paths don’t form a new cluster, indicating that the minimal changes to induce allodynia rely on both release and escape mechanisms.

These results suggest that when inhibitory gating of excitatory cell activity depends on the summed signaling from distinct inhibitory populations, the disruption of E-I balance is biased primarily towards reduced inhibitory activity (i.e. release mechanisms).

2.3 Analysis of the subcircuit mediating dynamic allodynia

In the dynamic subcircuit, allodynia is defined as firing of the E2 population in response to typically non-painful Aβ input. In this case, the APS is defined in the 7-dimensional space of coupling strength parameters (g_AβI1, g_I1E1, g_AβE1, g_E1E2, g_AβI2, g_I2E2, g_AβE2) and the allodynia surface is a hypersurface in this space. The structure of the dynamic subcircuit consists of two simple subcircuits coupled together, one consisting of E1 and I1, and the other consisting of E2 and I2 (Fig 6A). Our analysis shows that the most likely allodynia mechanisms exhibit similarities to the simple subcircuit, but the locus of release and escape mechanisms are distributed across different subcircuit components.

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Fig 6. Allowable parameter space (APS) for the proposed subcircuit mediating dynamic allodynia.

(A) A schematic of the dynamic subcircuit. I1 and I2 represent the populations of inhibitory neurons, and E1 and E2 represent the populations of excitatory neurons. We assume the E2 population relays signals to projection neurons. Aβ represents inputs to the subcircuit, relayed from the periphery along Aβ fibers. (B) The mean (black lines) and range (shaded gray areas) for the firing rate (top left) and voltage (top right) of the E2 population, as well as for the firing rates of the I2 (middle left), E1 (middle right), and I1 (bottom left) populations calculated across 20 sampled sets of coupling strengths each with a different random input Aβ stimulus (active during t ∈ [0.2, 0.7] s). (C) Violin plots of distributions of coupling strength values in the APS. The vertical bar represents the values within one standard deviation of the mean (square marker) for the corresponding coupling strength. (D) Normalized Pearson’s correlation coefficients between coupling strength values across sampled sets in the APS.

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2.3.1 The APS for the dynamic subcircuit.

We again define the APS by imposing conditions on the steady-state voltages of the E1, E2, I1, and I2 populations so the dynamic subcircuit displays experimentally-observed behaviors under normal healthy conditions (Table 3). To account for the phenomenon of pain inhibition, we require that steady-state voltages of both E1 and E2 are hyperpolarized from resting voltage by inhibitory input. However, if the E1 population is ablated, the E2 voltage remains within the reasonable bounds. Under ablation of the I1 population, we require that the E1 population fires in response to typical Aβ stimuli, and in turn excites the E2 population to fire. Finally, if the I2 population is ablated, then we require that the E2 population fires in response to typical Aβ stimuli. These conditions on the steady-state population voltages (Table 3) are re-written as a set of inequalities for the coupling strength values that must be satisfied for all f_Aβ input levels in the range [10, 20] Hz (see Section 4.4.3). The APS for the dynamic subcircuit is then defined as the sets of 7-tuples of coupling strengths (g_AβI1, g_I1E1, g_AβE1, g_E1E2, g_AβI2, g_I2E2, g_AβE2) which satisfy the system of inequalities and optimization problems.

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Table 3. Conditions that the proposed subcircuit mediating dynamic allodynia must satisfy and the resulting inequalities on steady-state voltages in response to Aβ input.

We ensure that the subcircuit exhibits desired behaviors by imposing conditions (middle column) on the subcircuit. Each condition is exhibited in either control, E1-ablation, I1-ablation, or I2-ablation conditions (left-most column), and is realized as an inequality (right-most column) on the steady-state voltage of a population. Inequalities must be satisfied for f_Aβ ∈ [10, 20] Hz, unless otherwise noted.

https://doi.org/10.1371/journal.pcbi.1012234.t003

Simulating 20 instantiations of the subcircuit with noisy Aβ input (Fig 6B), we observe high firing rates in the I1 and I2 populations that prevent sustained E1 and E2 firing activity and decrease E2 average voltage, as required for pain inhibition. Violin plots of the sampled APS points (Fig 6C) show that the coupling strengths governing the response of the populations to Aβ input, namely g_AβI1, g_AβE1, g_AβI2 and g_AβE2, have the largest ranges, reflecting low sensitivities of these parameters for obtaining normal healthy subcircuit responses. On the other hand, the coupling strengths from the inhibitory populations to the excitatory populations, namely g_I1E1 and g_I2E2, have the smallest ranges, suggesting that excitatory population responses to inhibitory signaling are more constrained to maintain healthy subcircuit responses.

APS points show high positive correlations between and , and between and (Fig 6D). This reflects that responses to inhibitory and excitatory inputs are balanced in the E1 and E2 populations, as similarly observed in the simple subcircuit. Also similar to the simple subcircuit, negative correlations between g_AβI1 and g_I1E1, and between g_AβI2 and g_I2E2, indicate a preservation of inhibitory signaling to each excitatory population such that weak activity in the inhibitory populations is compensated by higher sensitivity of the excitatory populations to their activity. Further, coupling strengths pertaining specifically to the I2-E2 component are only weakly correlated with coupling strengths pertaining to the I1-E1 component, indicating that excitation and inhibition are being balanced separately within each subcircuit component.

2.3.2 Mechanisms of allodynia in the dynamic subcircuit.

For this subcircuit, we consider the allodynia surface S_dyn as the set of points in normalized parameter space above which subcircuit instantiations produce E2 firing for at least some typical f_Aβ (see Section 4.7). Clustering points of the APS according to the directions of their shortest vectors to S_dyn (Sections 4.8 and 4.9) yields four clusters (Fig 7A). The APS points in Cluster 1 (green) are characterized by lower values and higher values while the APS points in Cluster 2 (blue) exhibit higher values, compared to the mean of the APS. The APS points in Cluster 3 (red) are characterized by lower and higher and values, and the points in Cluster 4 (cyan) exhibit higher and values, compared to the mean of the APS. All four clusters are similarly distanced from the allodynia surface, with Cluster 1 being slightly closer (Fig 7B). Paths from APS points to the allodynia surface whose distances were also local minima (as opposed to the global minima for each point) fall into four similar clusters, indicating the robustness of the shortest path directions (S1 Fig, panel C).

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Fig 7. Mechanisms for dynamic allodynia represented as shortest paths to the allodynia surface for the dynamic allodynia subcircuit.

(A) Violin plots for coupling strength distributions for the APS points in the 4 clusters of directions of shortest path lengths to the dynamic allodynia surface: Cluster 1 (green, top panel), Cluster 2 (blue, 2nd panel), Cluster 3 (red, 3rd panel), and Cluster 4 (cyan, bottom panel). Black * represents the mean APS coupling strength value, colored square is mean value for the cluster. (B) Probability distribution of the shortest distances to the dynamic allodynia surface (overall profile); shading represents the contributions of each cluster to the overall profile. For instance, a bar that is 50% dark blue indicates that Cluster 2 constitutes 50% of subcircuit instantiations with the corresponding distance to the allodynia surface. Only ∼2% of sampled subcircuit instantiations belong to Cluster 4 with distances greater than 3.5, thus contributing minimally to distribution. (C) Parallel plot representation of the directions of the shortest paths to the allodynia surface from each sampled point in the APS. A line gives the components of the shortest displacement vector from an APS point to the allodynia surface. Lines are colored based on clusters with a color gradient for better visualization. (D) Firing-rate responses to a noisy Aβ input signal (active during t ∈ [0.2, 0.7]) for each population (columns) and for each cluster (rows). Solid lines correspond to a subcircuit instantiation with coupling strengths given by the mean APS values for the particular cluster, and dash-dotted lines correspond to the subcircuit instantiation with coupling strengths given by the corresponding closest point on the allodynia surface. Aβ input frequencies are chosen as the smallest value that induces allodynia for each cluster.

https://doi.org/10.1371/journal.pcbi.1012234.g007

In terms of mechanisms for producing allodynia, Clusters 1 and 3 primarily represent the release from inhibition allodynia mechanism in the I2-E2 component and I1-E1 component of the subcircuit, respectively. For either cluster, reaching the allodynia surface requires a decrease in the effect of Aβ input to the inhibitory population and a smaller increase in the effect of Aβ input to the excitatory population (green and red curves in Fig 7C). This results in either excitatory population primarily being released from its inhibitory control with a weaker escape effect, as is evident in example subcircuit responses to noisy Aβ input (Fig 7D, first and third rows). Note that for Cluster 3, releasing the E1 population from inhibitory control provides excitatory input to the E2 population that allows it to escape the inhibitory control provided by the I2 population.

In contrast, Clusters 2 and 4 represent escape from inhibition in each subcircuit component, respectively. Allodynia is induced in these subcircuit instantiations by decreasing the response of the excitatory population to inhibitory input while simultaneously increasing the effect of the Aβ input to the excitatory population (see blue and cyan curves in Fig 7C). This results in the E2 population escaping the inhibitory control from the I2 population (Cluster 2) and the E1 population escaping the control of the I1 population (Cluster 4). Again, in Cluster 4, the escape of the E1 population provides excitatory input to the E2 population to allow it to escape its inhibitory control. We note that Cluster 4 comprises the smallest portion of the APS (∼2%) and its points are significantly farther from the allodynia surface than the remainder of the APS.

Just as the structure of the dynamic subcircuit shows symmetry relative to the simple subcircuit, the most likely mechanisms for allodynia are also similar with escape and release from inhibition basically equally likely to occur across the APS. However, the locus for escaping inhibition is primarily the E2 population (Cluster 2, Fig 7B). The remaining half of the APS has mechanisms split between E2 release from inhibition (Cluster 1) and E1 release from inhibition (Cluster 3). For the dynamic subcircuit, these results suggest that while the presence of E1 can act as an amplifier of excitatory Aβ signaling to E2, it may not be the most likely culprit in tipping E-I balance towards excitation to cause allodynia.

3.1 Application to more complex dorsal horn circuits

These results can be applied to help understand how E-I balance is maintained and likely disrupted in more complex dorsal horn circuits [34, 35]. In models of more complex circuits, the space of unconstrained parameters is so high-dimensional that identifying the diversity of parameter sets that satisfy desired model behaviors is difficult, even when computational optimization algorithms are implemented. Our results can provide constraints on regions of parameter space where E-I balance in subcircuits of the network is achieved. For example, the dorsal horn laminae I-III neural circuit modeled by Medlock et al. [34] consists of 5 excitatory interneuron populations and two inhibitory interneuron populations with each population consisting of a network of Hodgkin-Huxley-type model neurons. In the circuit, 3 of the excitatory populations act as upstream amplifiers of Aβ input to 2 downstream excitatory populations that make direct connections to projection neurons. The synaptic pathways of Aβ input through the upstream excitatory populations to one of the downstream excitatory populations are similar to the dynamic subcircuit modeled here. Our results predict that there should be parameter sets that limit responses to Aβ input in the downstream excitatory population through these synaptic pathways by mechanisms reflected by the clusters found for the dynamic subcircuit. Namely, there should be parameter sets in which the downstream excitatory neuron responses are primarily gated by direct inhibition from the inhibitory cells targeting them (corresponding to Clusters 1 and 2), and parameter sets in which their responses are limited by inhibitory control of the upstream excitatory populations (corresponding to Clusters 3 and 4). Based on Medlock et al.’s [34] finding that, in order to maintain healthy E-I balance, small reductions in inhibitory control of upstream excitatory populations required larger increases in inhibitory input to the downstream excitatory population suggests that perhaps their model parameter set may be analogous to Cluster 3 or 4 parameter sets in the dynamic subcircuit. However, since the Medlock et al. [34] network contains additional components, there may be additional constraints on achieving E-I balance that are not contained in the smaller dynamic subcircuit.

3.2 Model limitations

A limitation of the firing rate model formalism we implemented, which only models average population voltages and firing-rates, is that specific excitability characteristics or response features that are evident on the single neuron and spike levels may not be captured by the sigmoidal steady state firing rate activation functions. Nevertheless, we did constrain the dependence of population average firing rates on membrane voltages by fitting the steady state activation functions to frequency-voltage relationships measured in dorsal horn neurons [26]. However, these relationships do not take into account some spiking patterns observed in dorsal horn excitatory interneurons, such as delayed onset of firing or transient firing [16, 20, 36, 37]. Recent development of next-generation firing rate models that can be directly reduced from networks of individual neuron models [38, 39] provide the framework to include specific spiking patterns into a mean-field reduction, such as spike frequency adaptation [40, 41]. We expect that delayed firing could similarly be accounted for in a firing-rate model reduction of simplified neuron models that are fit to observed firing patterns of dorsal horn cells. Including such spiking properties in a firing rate population network would allow analysis of the interactions of these cellular firing patterns with network structure in maintaining or disrupting E-I balance in dorsal horn circuits.

Experimental studies have shown that static and dynamic allodynia are mediated by different afferent inputs [42–44]. Specifically, dynamic allodynia is mediated by Aβ fiber input which in healthy conditions is unable to activate laminae I-II nociceptive circuits, while static allodynia can be mediated by Aβ fiber input or by sensitized mechanical nociceptors (Aδ or C fiber input) depending on the pathological or experimental conditions [42]. Our analysis focused on dysregulation of inhibitory gating of Aβ fiber input for both allodynia types. Further experiments are needed, however, to identify if signaling patterns on Aβ fibers differ in response to brushing (dynamic) or punctate (static) stimuli.

Our models did not include activity on nociceptive Aδ or C fibers, or responses of lamina I projection neurons. The model subcircuits could be extended to explicitly include the effects of painful signals arriving on Aδ/C fibers that make direct connections on projection neurons and interneuron populations. In previous work modeling spinal subcircuits using a firing rate model formalism, we included C fiber input that was mediated through NMDA receptors in post-synaptic populations [45]. The NMDA-receptor mediated connection strength depended on post-synaptic average voltage to account for voltage-dependent removal of a Mg²⁺ block. Such an extended model would be able to account for wind-up of projection neuron activity in response to repetitive brief C fiber input and also explicitly account for pain inhibition, namely the attenuation of C fiber response in the projection neurons in the presence of simultaneous Aβ input. We note, however, that our current results would not be qualitatively affected as Aβ and C fibers are parallel pathways and their inputs do not interact.

3.3 Application of methodology

In this study, we introduced an analysis methodology for neural population model circuits that can determine parameter values optimized to account for experimental observations and for identifying sensitivities of model behaviors to parameter variations. The methodology involves the following steps:

1. Translate normal and pathological experimental behaviors that a circuit should replicate into analytical conditions that model variables must satisfy.
2. Re-frame these conditions into systems of inequalities and optimization problems that the parameters of interest must satisfy. In our work, we focused on the parameters governing the coupling strengths between populations. These analytically determined conditions described distinct regions of parameter space in which the corresponding subcircuit instantiations displayed normal behaviors (the APS) and above which they displayed pathological behavior (above the allodynia surface).
3. Determine the most likely mechanisms that induce the pathological condition, in our case allodynia, by finding the shortest path from each parameter set in the APS to the surface at the boundary of the pathological region

This methodology can be applied to study any circuit of neural populations, although the implementation of Step 2 is more tractable for feedforward circuits with a relatively small number of populations. Thus, we expect that the methodology may be useful for analyzing propagation of signaling and E-I balance in other sensory processing circuits with feedforward structure.

3.4 Conclusions

In all, successful treatment of chronic pain conditions, such as allodynia, require full understanding of the underlying physiological causes. Dysregulation of E-I balance in dorsal horn circuitry is a compelling cause supported by an array of pre-clinical studies, but our incomplete understanding of the circuitry and the building evidence for its complexity leave many questions for how the dysregulation occurs. The identification of multiple types of excitatory and inhibitory interneurons in dorsal horn circuits suggests that dysregulation may occur through multiple mechanisms that may be dependent on the nature of the injury or insult that induces the chronic pain condition [46]. Our modeling results are a first step to systematically unravel the interactions among multiple interneuron populations for the maintenance of E-I balance in healthy conditions and its disruption in allodynia. Our identification of a wide diversity of mechanisms underlying allodynia in simplified dorsal horn subcircuits suggest potential sources for the high interindividual variability observed in allodynia conditions. Continued experimental work identifying dorsal horn circuit structure and the functional relationships among diverse interneuron types will provide constraints necessary to improve models of spinal circuitry [47] so that they can better participate in the development of therapies for this debilitating condition.

4.1 Population firing rate model

For our models of laminae I-II dorsal horn neuronal subcircuits, we implement a well-established firing rate model formalism that models the average membrane voltage and average firing rates of neuronal populations (see e.g. [23–25]). Average firing rates f_x are computed from average voltages V_x with a sigmoidal activation function of the form: (2) where m_x is the maximum firing rate of the population, β_x is the half-activation voltage and α_x governs the slope of the population’s firing-rate response to voltage changes. We match these parameters to experimental measurements of frequency-voltage relationships for dorsal horn neurons as described in Section 4.2. We expect that in the absence of inputs, V_x remains at a rest value V_x,rest. In response to synaptic inputs, V_x deviates from its rest value according to the following differential equation: (3) where the time constant τ_x describes how quickly V_x changes in response to inputs. The inputs are computed as the sum of the firing rates of all pre-synaptic populations to population x, denoted here as y₁, y₂, …, together with the firing rate of the Aβ fiber input, f_Aβ, weighted by the corresponding coupling strengths:

We thus expect that in the presence of steady inputs, V_x approaches the steady-state value given by

Table 4 summarizes the model equations for each subcircuit we analyze.

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Table 4. Equations for each subcircuit model.

https://doi.org/10.1371/journal.pcbi.1012234.t004

4.2 Parameters of population firing-rate models

We choose parameters for the activation functions of excitatory and inhibitory populations based on the experimental measurements of membrane properties and firing behavior in rat dorsal horn neurons reported in Ruscheweyh et al. [26]. In our subcircuits, we assume all excitatory populations have the same parameters and assume the same for inhibitory populations. We assume that average resting voltages V_I,rest and V_E,rest are −60 mV, approximately the values reported in [26]. Maximum firing rates of excitatory and inhibitory populations are set to 50 and 80 Hz respectively, based on [34].

We use the frequency-current relations and average voltage-current relations reported in [26] to extract frequency-voltage relations. Specifically, frequency and average voltage values for the same applied current levels were estimated from graphs of the two relations. The firing-rate activation functions given in Eq (2) are fit to these frequency-voltage relations using the trust-region-reflective non-linear-least-squares algorithm via Matlab’s “fit” function [48] to obtain the values of β_x and α_x (x = E, I) for excitatory and inhibitory populations (Fig 8). Inhibitory population relations were fit to data from lamina I fast tonic firing neurons and excitatory population relations were fit to data from lamina I and II delayed firing neurons.

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Fig 8. Frequency vs. voltage functions.

Frequency vs. voltage relations (curves) were fit to frequency vs. voltage data (circles, [26]) for (A) lamina I tonically-firing neurons for inhibitory populations and (B) for lamina I and II delayed-firing neurons for excitatory populations.

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Maximum and minimum voltages are set to 12α_x mV above and below, respectively, the half-activation voltage value β_x. The voltage thresholds for firing are defined as β_x−α_x. Choosing voltage thresholds and maximum and minimum voltages in terms of α_x and β_x ensures that both populations have similar behavior in response to proportional changes in input. In particular, these choices ensure that at the minimum voltage, the maximum voltage, and the voltage threshold, firing rates for both populations will be roughly 4⋅10⁻¹¹ ≈ 0, (1 − 4⋅10⁻¹¹) ≈ 1, and 0.1 of their maximum firing-rates, respectively.

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Table 5. Model parameter values for inhibitory and excitatory populations.

Parameters for each population type (first column) are divided into 3 categories (2^nd column): those that pertain to activation functions, those that give voltage cutoffs, and those that appear in the model differential equations. All these parameter values are fixed for each inhibitory or excitatory population.

https://doi.org/10.1371/journal.pcbi.1012234.t005

Finally, we choose membrane time constants so they are roughly on the same time-scale as those of [49], and the ratio of τ_E to τ_I is 1.2 as cited in [26]. Notably, since many of the results depend on steady-state voltages, small changes in the particular values of the time constants τ_E and τ_I have little effect on the qualitative behavior of the results. We choose τ_E = 0.024 seconds and τ_I = 0.02 seconds. All model parameters are listed in Table 5.

4.3 Aβ stimuli model

In simulations of subcircuit responses to time-varying input on Aβ fibers, we drive neural populations with input representing the average firing rate of a bundle of 300 Aβ fibers (which is on the order of magnitude of the number of Aβ fibers in afferent nerve fibers from rat skeletal muscle, as in e.g [50]). In particular, we describe the spiking activity on each Aβ-fiber in the bundle via a Poisson process with a variable rate depending on the presence or absence of peripheral stimuli. Namely, in the absence of stimuli, each fiber transmits action potentials at a background rate of 1 Hz, while in the presence of a stimulus the rate increases to f_Aβ Hz. The input to subcircuit populations is the average spiking rate across all Aβ fibers (in Hz), namely a noisy time-varying input with average firing rate f_Aβ Hz. In all subcircuit simulations, stimuli are applied from 0.2 − 0.7 seconds.

In the parameter sensitivity analysis of model subcircuits, f_Aβ is taken as a constant value between [10, 20] Hz. This range of Aβ fiber firing activity has been observed in slowly adapting mechanoreceptors in response to ramp-and-hold mechanical stimulation [27].

4.4 Defining the allowable parameter space (APS) for the subcircuits

Our parameter sensitivity analysis method consists of translating normal experimental behaviors that the subcircuits should replicate into analytical conditions on model variables, specifically on average voltages. These conditions are then re-written into systems of inequalities and optimization problems that coupling strength parameters must satisfy. The parameter sets that satisfy these systems constitute the allowable parameter space (APS). In this section, we derive these systems for our model subcircuits. Full details of the derivation are shown for the simple subcircuit only, as the derivation follows similarly for the static and dynamic subcircuits.

4.4.1 Simple subcircuit.

For the simple subcircuit, it is possible to make considerable progress towards deriving an explicit definition of the allowable parameter space. As described in Table 1 in Section 2.1.1, the conditions on average voltages that ensure the subcircuit replicates experimentally appropriate behaviors are given as follows: (4)

The first two inequalities can be rewritten to yield bounds on g_AβI:

However, as these conditions must hold for all f_Aβ ∈ [10, 20] Hz, we need that which can be re-written as

Likewise, the last two inequalities in Eq (4) yield analogous bounds on g_AβE and the middle two inequalities yield analogous bounds also on g_AβE. We summarize the resulting system of inequalities for coupling strength parameters in Table 6.

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Table 6. Inequalities on coupling strengths that define the allowable parameter space for the simple subcircuit.

Inequalities on simple subcircuit coupling strength parameters that define the subcircuit APS and are obtained from the inequalities on population voltages in Table 1. (UB = upper bound, LB = lower bound, abl = ablation, = does not fire above threshold levels, PI = pain inhibition).

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Since f_I is a nonlinear (hyperbolic tangent) function of f_Aβ and g_AβI, the optimization problems in the last line of Table 6 are generally difficult to solve explicitly. Nevertheless, we can simplify the maximization problem——and in fact explicitly solve the minimzation problem——making it much easier to numerically approximate the APS. To do so, note that both of those optimization problems can be rewritten as: where

Consequently, the solutions to the preceding optimization problems occur either at x = g_AβI f_Aβ,min/α_I, g_AβI f_Aβ,max/α_I, or at one of the critical points given by (5)

We show in Section A in S1 Appendix that when C = 0, the solutions of Eq 5 are given in terms of the 0^th, W₀, and −1^st, W₋₁, branches of the Lambert-W function: (6) which yields so long as f_Aβ0 or f_Aβ−1 ∈ [f_Aβ,min, f_Aβ,max] or at f_Aβ = f_Aβ,min or f_Aβ = f_Aβ,max.

In Section A in S1 Appendix, we address the case when C ≠ 0 by showing how to take advantage of the structure of the optimization problem to solve it numerically.

As an alternative approach, it is possible to address the case where C ≠ 0 by rewriting the middle three equations of Eq 4 so they bound g_IE (see Section B in S1 Appendix). The resulting optimization problems may then be solved explicitly in terms of the Lambert W functions. While we do not implement this alternative approach for the simple circuit, we do apply it in our analysis of the static subcircuit (see below).

4.4.2 Static subcircuit.

For the static subcircuit, the inequalities that population voltages must satisfy to replicate experimentally-observed behaviors are listed in Table 2 in Section 2.2.1. Following a similar derivation as for the simple subcircuit, we rewrite the conditions as the system of inequalities and optimization problems on coupling strength parameters given in Table 7. The inequalities in the last four rows of Table 7 involve maximizing or minimizing a quantity over the range of f_Aβ values. To make the APS easier to compute, we find explicit solutions to the four optimization problems in the last two rows of Table 7 using the alternative approach described above for the simple subcircuit and outlined in Sections B and C in S1 Appendix. The solutions to the four optimization problems (Table 8, 3rd column) can be written in terms of the Lambert W₀ function (4th column) with different constants A (5th column).

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Table 7. Inequalities expressed as upper and lower bounds on coupling strengths define the allowable parameter space for the static subcircuit.

These inequalities are obtained by algebraically manipulating the inequalities on the voltages of various populations from Table 2 that define the APS for the static subcircuit so that the inequalities are written explicitly in terms of coupling strengths. (UB = upper bound, = does not fire above threshold levels, LB = lower bound, abl = ablation, PI = pain inhibition).

https://doi.org/10.1371/journal.pcbi.1012234.t007

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Table 8. Explicit solutions of f_Aβ for optimization problems in Table 7.

The value of f_Aβ at extrema satisfying the four optimization problems in the last two rows of Table 7 (3rd column). All solutions (4th column) have the same form involving the Lambert W₀ function with different constants A (5th column) (UB = upper bound, abl = ablation).

https://doi.org/10.1371/journal.pcbi.1012234.t008

To more easily sample from the APS, it is helpful to compute bounds on E population coupling strength values that are independent of other coupling strength parameters. For example, to find an upper bound on g_I1E, we can use the upper bound on E during I1-ablation, (rewritten so the inequality is expressed as bounds on g_AβE rather than g_I1E), along with the E lower bound given no ablations to obtain that which requires in particular that for all f_Aβ ∈ [f_Aβ,min, f_Aβ,max] which in turn implies that

To find a lower bound on g_I1E, on the other hand, we can use the E lower bound under I1 ablation, (rewritten so the inequality is expressed as bounds on g_AβE rather than g_I1E), along with the pain inhibition condition, i.e. that: which, via an argument analogous to the preceding argument used to find an upper bound on g_I1E, implies that

An analogous procedure produces bounds on g_I2E.

4.4.3 Dynamic subcircuit.

We rewrite the inequalities in Table 3 in Section 2.3.1 as the system of inequalities and optimization problems for the coupling strength parameters given in Table 9.

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Table 9. Inequalities expressed as upper and lower bounds on coupling strength values define the allowable parameter space for the dynamic subcircuit.

These inequalities are obtained by algebraically manipulating the inequalities on the voltages of various populations from (Table 3) that define the APS for the dynamic subcircuit so that the inequalities are written explicitly in terms of coupling strength parameters. (UB = upper bound, LB = lower bound, abl = ablation, PI = pain inhibition, = does not fire above threshold levels).

https://doi.org/10.1371/journal.pcbi.1012234.t009

We further simplify the inequalities in the second line of Table 9 by explicitly solving the optimization problems in terms of Lambert-W functions analogously to our treatment of the optimization problems in the last row of Table 6 for the simple subcircuit. Upper and lower bounds on g_I1E1 are straightforward to find, because the I1-E1 portion of the dynamic subcircuit has identical constraints to those of the simple subcircuit. To find upper and lower bounds on g_I2E2, we use a more computationally intensive approach. Namely, given the parameter values for the I1-E1 portion of the subcircuit and given g_E1E2, we find the set of g_I2E2 values such that the inequalities in the bottom-most three lines of Table 9 have a solution. That is, we choose g_I2E2 so that the upper bounds on g_AβE2 are indeed larger than the lower bounds appearing in the last three lines of Table 9.

4.5 Normalizing the allowable parameter space (APS)

We normalize the APS so that changes in different coupling strength parameters can be compared. Since normalization requires upper and lower bounds on each coupling strength parameter, we construct a rectangular hypercube in parameter space that contains the APS. To do this, we leverage the hierarchical nature of the sets of inequalities on coupling strength parameters for each subcircuit in Tables 6, 7 and 9. The hierarchy is formed by the dependencies of inequalities on the coupling strength parameters, with inequalities higher in the hierarchy depending on parameters defined by inequalities lower in the hierarchy. Table 10 lists coupling strength parameters for each subcircuit in the order of the hierarchy formed by their inequalities (from low to high).

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Table 10. Hierarchical order formed by the inequalities on coupling strength parameters for each subcircuit.

Order of the hierarchy formed by the inequalities for coupling strength parameters shown in Tables 6, 7 and 9 listed from lowest to highest in the hierarchy.

https://doi.org/10.1371/journal.pcbi.1012234.t010

Here we describe the algorithm implemented to construct the rectangular hypercube in parameter space that contains the APS based on the hierarchy of inequalities on coupling strength parameter values. In general, consider a hierarchical set of inequalities that define bounds on elements of a parameter vector in :

• a₁ ≤ x₁ ≤ b₁
• L₂(x₁)≤x₂ ≤ U₂(x₁) for x₂ ∈ [a₂, b₂]
• L₃(x₁, x₂)≤x₃ ≤ U₃(x₁, x₂) for x₃ ∈ [a₃, b₃]
  ⋮
• L_n(x₁, x₂, …, x_n−1)≤x_n ≤ U_n(x₁, x₂, …, x_n−1) for x_n ∈ [a_n, b_n]

for real numbers a₁ ≤ b₁, …a_n ≤ b_n, and real functionals L₁ ≤ U₁, …, L_n ≤ U_n. As written, the x_n inequalities are the highest in the hierarchy because they depend on x₁, …, x_n−1. To identify the interval of values [x_i,min, x_i,max] that satisfies the inequality for each x_i, we generate a large (at least 1000 elements), random (not necessarily uniform) sample of values satisfying the inequality system as follows:

• Uniformly at random choose a value of x₁ in [a₁, b₁]
• Given the value of x₁, uniformly at random choose a value of x₂ ∈ [L₂(x₁), U₂(x₁)] if such an interval exists. If such an interval doesn’t exist, start over.
• Repeat to choose x₃, …, x_n sequentially. If, at any step, an interval for x_i is not defined, start over with a new choice for x₁.

We define the minimum (maximum) value of each coupling strength parameter in the APS as the minimum (maximum) value across all samples. This defines a hypercube in that contains the APS. To normalize, we map each element of to [0, 1] as follows:

The normalized APS thus lies in the unit hypercube.

4.6 Sampling from the allowable parameter space (APS)

To perform analyses, we generate a uniform sampling of the APS. However, the APS is high-dimensional, can be non-convex, and is generally of an unknown shape, making it difficult to sample uniformly. In this section we briefly outline the algorithm we developed to sample the APS for each subcircuit; further details are contained in S2 Appendix.

Having found a rectangular subspace that contains the APS (Section 4.5), one approach would be to uniformly at random sample a point from the rectangular subspace, check to see if the point is in the APS, and keep it if it is. However, this naive sampling scheme is dependent on the volume of the APS in such a way that its computational complexity is exponential in n (see Section B in S2 Appendix).

4.6.1 Volume-independent sampling algorithm.

Here we describe a spatially uniform sampling algorithm whose computational complexity is independent of the volume of the APS in Rⁿ. To do so, we sample from a cover of the normalized APS consisting of Rⁿ hyperrectangles that approximates the normalized APS. Our algorithm takes the following strategy:

1. Define a set of Rⁿ hyperrectangles that contains and approximates the allowable parameter space.
2. Uniformly at random select a hyperrectangle
3. Uniformly at random select a point from the hyperrectangle.
   • If the point is not in the parameter space, discard it
4. Otherwise, keep the point with probability proportional to the volume of the hyperrectangle.

Step (4) ensures that the sample is indeed uniform-in-space. Fig 9 provides an illustration of the sampling algorithm. (See S2 Appendix for details on the implementation of this algorithm, a discussion of the volume-independence of the implementation, and proof that this produces a uniform in space sampling).

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Fig 9. Schematic of the volume-independent sampling algorithm.

https://doi.org/10.1371/journal.pcbi.1012234.g009

4.7 Defining the allodynia surface

In this section, we derive the conditions describing the allodynia surface for each subcircuit. We present the derivation in terms of a generalized circuit and conditions for a general target state to occur.

In particular, we restrict our attention to circuits where the output signal is relayed by a single neural population, which we denote by y. The target state is represented by the voltage of this output population V_y increasing above a specified threshold in response to an input signal. For our firing rate model, the steady-state average voltage V_y is given by where f_in is the input signal and are firing rates of the populations pre-synaptic to population y. The vector contains all the coupling strength parameters in the circuit, g_ji for the weight of the connection from pre-synaptic population j to postsynaptic population i. The circuit is in the target state when

In our subcircuits, the output populations y are the excitatory interneuron populations that directly target the projection neurons that relay signals to the brain, namely the E, E, and E2 populations in the simple, static and dynamic subcircuits, respectively. The target state for allodynia is that the average voltage of these excitatory populations increases above the firing threshold in response to Aβ input. These allodynia conditions are summarized in Table 11.

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Table 11. Allodynia conditions on each subcircuit.

Allodynia conditions specifying precisely when the subcircuit is producing allodynia–when it is relaying pain-inducing stimuli in response to innocuous f_Aβ signals.

https://doi.org/10.1371/journal.pcbi.1012234.t011

Using the target state condition, we can identify the coupling strength values for which the target state is attainable. To do so, we rewrite the condition defining the target state as an inequality on the coupling strength g_in,y between the input signal and the population y: (7) (8)

For the target state to be attainable, we don’t need to reach the target state for all values of f_in. Instead, we can reach the target state for the value of f_in that minimizes the right-hand side of the preceding equation. Thus, the circuit with the set of coupling strengths can attain the target state if and only if (9)

To illustrate this more concretely, the conditions on the coupling strengths for which allodynia is attainable are summarized in Table 12 for the simple, dynamic, and static subcircuits.

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Table 12. Conditions for a subcircuit to produce allodynia.

These conditions specify the sets of coupling strength values when the corresponding subcircuit will produce allodynia for some typical f_Aβ input.

https://doi.org/10.1371/journal.pcbi.1012234.t012

Eq (9) thus defines a target state boundary surface S which divides the sets of coupling strength values for which the circuit is in the target state from those for which it is not. We can express this surface as (10)

For the simple, static, and dynamic subcircuits, this boundary (Table 13) separates regions of parameter space in which the subcircuit can produce allodynia from regions where it does not.

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Table 13. The allodynia surface for the simple, dynamic and static subcircuits, respectively.

The allodynia surface separates subcircuit instantiations which can produce allodynia in response to typical f_Aβ signaling from those that cannot.

https://doi.org/10.1371/journal.pcbi.1012234.t013

4.8 Computing the distance between sampled points and the allodynia surface

In this section, we discuss the computational algorithm that computes the shortest path from points in the APS to the allodynia surface. Similarly as in Section 4.7, we describe the algorithm for a generalized circuit and a generalized target state boundary surface S.

The length of the shortest path between a point in the space of coupling strength values to the target state boundary surface S indicates how easy it is to move the circuit into the target state. It also identifies which coupling strengths need to change to reach the target state. We illustrate the problem of finding the shortest path to S by considering an arbitrary set of coupling strength values–a point in the APS which we denote . To compute the shortest path from to S, we need to find the point on S closest to . To do so, we need to solve the optimization problem: (11) where ||⋅|| represents the Euclidean norm.

Solving the optimization problem in Eq (11) directly would require knowing the target state surface itself or knowing important properties of it such as its gradient. However, the target state boundary surface in our work is defined via solving a minimization problem over the space of coupling strength values (excluding the coupling strength g_in,y) and is thus difficult to include in existing optimization algorithms. Thus, we seek to solve this problem without computing an explicit representation of the target state surface S.

Instead, we solve the higher dimensional problem (12) where S* is the following set of coupling strength-input signal pairs defined by removing the minimization from Eq (10):

Eq (12) presents a constrained optimization problem that can be solved with high likelihood using global optimization algorithms based on stochastic gradient descent. As a result, Eq (12) is far more tractable than the original problem posed in Eq (11). Moreover, if , the two minimization problems are equivalent, and by solving Eq (12), we will have solved the original minimization problem Eq (11). In S3 Appendix, we describe how we solve Eq (12) using Matlab’s stochastic gradient descent-based algorithm fmincon in a multi-start global optimization scheme and show that if we solve Eq (12), we do indeed solve Eq (11).

4.9 Clustering the data based on shortest paths to the allodynia surface

We use a density-based scanning clustering algorithm coupled with data visualization to identify clusters in the APS. To do so, we work with the uniformly sampled points in the APS and cluster them according to their shortest paths to the allodynia surface.

To apply density-based clustering to the data, we use Matlab’s dbscan function. Briefly, dbscan divides the data into equivalence classes, where two datapoints are equivalent if they are sufficiently close, and identifies equivalence classes as a cluster if they contain a point which exceeds a minimum number of sufficiently close neighbors. The function takes three arguments:

1. The data to be clustered: We use the set of shortest path vectors from sampled points in the APS to the allodynia surface.
2. A sufficiently close distance ϵ: we use the smallest ϵ under the euclidean metric that leads to no outliers in the data.
3. The minimum number of sufficiently close neighbors to identify an equivalence class as a cluster: we take this to be 5.

Figs 3D, 5C and 7C show parallel plots of the shortest paths to the allodynia surface for the simple, static, and dynamic subcircuits, respectively; and are colored according to the clusters assigned by density-based scanning. The shortest paths in those figures clearly divide visually into spatially-separated clusters.