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. 2008 Jan;4(1):e11.
doi: 10.1371/journal.pcbi.0040011. Epub 2007 Dec 13.

Neuronal firing sensitivity to morphologic and active membrane parameters

Christina M Weaver  1 , Susan L Wearne
Affiliations

Neuronal firing sensitivity to morphologic and active membrane parameters

Christina M Weaver et al. PLoS Comput Biol. 2008 Jan.

Abstract

Both the excitability of a neuron's membrane, driven by active ion channels, and dendritic morphology contribute to neuronal firing dynamics, but the relative importance and interactions between these features remain poorly understood. Recent modeling studies have shown that different combinations of active conductances can evoke similar firing patterns, but have neglected how morphology might contribute to homeostasis. Parameterizing the morphology of a cylindrical dendrite, we introduce a novel application of mathematical sensitivity analysis that quantifies how dendritic length, diameter, and surface area influence neuronal firing, and compares these effects directly against those of active parameters. The method was applied to a model of neurons from goldfish Area II. These neurons exhibit, and likely contribute to, persistent activity in eye velocity storage, a simple model of working memory. We introduce sensitivity landscapes, defined by local sensitivity analyses of firing rate and gain to each parameter, performed globally across the parameter space. Principal directions over which sensitivity to all parameters varied most revealed intrinsic currents that most controlled model output. We found domains where different groups of parameters had the highest sensitivities, suggesting that interactions within each group shaped firing behaviors within each specific domain. Application of our method, and its characterization of which models were sensitive to general morphologic features, will lead to advances in understanding how realistic morphology participates in functional homeostasis. Significantly, we can predict which active conductances, and how many of them, will compensate for a given age- or development-related structural change, or will offset a morphologic perturbation resulting from trauma or neurodegenerative disorder, to restore normal function. Our method can be adapted to analyze any computational model. Thus, sensitivity landscapes, and the quantitative predictions they provide, can give new insight into mechanisms of homeostasis in any biological system.

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Conflict of interest statement

Competing interests. The authors have declared that no competing interests exist.

Figures

Figure 1
Figure 1. Constructing the Morphologic Model
(A) Morphology of an Area II neuron traced in 3-D. The reduced morphology in (B) conserved the surface area of the soma (shown in green) and the length and surface area of the dendritic tree. The axon (truncated thick process extending from the left of the soma) was omitted. (B) Channel distributions in active and passive dendrite models. Red compartments included active channels; gray compartments included only passive ones. (C) Morphologic perturbations. Top left: unperturbed model morphology. Top right, L + SA: dendritic length L and surface area SA perturbed with dendritic diameter D held constant. Bottom left, D + SA: D and SA perturbed with L held constant. Bottom right, L + D: L increased and D decreased such that SA remained constant. (D) When dendritic morphology of the active dendrite was perturbed (e.g., L + SA), either the original dendritic channel density of each ion species was conserved (constant density; middle) or the number of dendritic channels was conserved (constant numbers; bottom). (E) Top, spatial distribution of active channels in the active dendrite. Dendritic channel density (as a proportion of its somatic density) either increased ([Image: see text] , thick dashed line), decreased ([Image: see text] , solid black line; [Image: see text] , [Image: see text] , [Image: see text] , red solid line), or remained constant ([Image: see text] , thin dashed line). See Materials and Methods. Bottom, decremental AP backpropagation as a function of distance from the soma, for the conductance distributions shown at top. Somatic AP shown in red.
Figure 2
Figure 2. Quantifying Model Output
(A) Computing firing rate gain. Black circles and solid line show firing rates and the associated fit for a model with a gain of 331.1 Hz/nA. A model with a gain of 640.0 Hz/nA is also shown (blue open triangles and dashed line). (B) Typical match of optimized model (red) against target data (black dashed line), for 0 and 100 pA injected current (top and bottom, respectively). Models were constrained to match the general AP shape, and approximate firing rates, of the synthetic target data.
Figure 3
Figure 3. Evaluating Sensitivity for Optimized Models Across Parameter Space
(A) Location in nine-dimensional parameter space of the 15 optimized passive dendrite models (gray circles and colored triangles), shown in three 3-D subspaces: [[Image: see text] , [Image: see text] , [Image: see text] ] (top); [Kp, RCa, [Image: see text] ] (middle); and [[Image: see text] , [Image: see text] , [Image: see text] ] (bottom). Voltage traces and sensitivity coefficients of three models represented as colored triangles (A, B, and C) are compared in (B), (C), and Figure 4. (B) Comparison of morphologic (D + SA) and active parameter ([Image: see text] ) perturbations of the model labeled “B” (blue triangles in [A], above). Solid blue line represents the somatic voltage trace of the unperturbed model. Dashed line shows the response to a 20% decrease in D + SA; solid black line shows the response to a 20% increase in [Image: see text] . Inset barplot shows the normalized sensitivity coefficients of spontaneous firing rate to D + SA and [Image: see text] (Sp. FR Sens.): positive to [Image: see text] , negative and of greater magnitude to D + SA. (C) Perturbations of the model labeled “C” (red triangles in [A], above). Shown are perturbations analogous to those in (B) above, except that the solid black line shows the response to a 10% increase in [Image: see text] . Sensitivity to D + SA is small and negative, while sensitivity to [Image: see text] is large and positive. Note the difference in scale between the sensitivity barplot insets in (B) and (C).
Figure 4
Figure 4. Sensitivity of Spontaneous Firing Rate to Model Parameters Varied Across Parameter Space
(A–C) Somatic voltage traces and normalized sensitivity coefficients to perturbations of active ([Image: see text] , [Image: see text] , [Image: see text] , Kp, RCa, [Image: see text] , [Image: see text] , [Image: see text] ) and morphologic (L + D, L + SA, D + SA) parameters, for the models labeled A, B, and C in Figures 3A and (D) (colored triangles). For some regions of parameter space (points A and B), the pattern of parameter sensitivities was similar. In other regions (C), the sensitivity to several parameters changed dramatically (compare colored bars in [A] and [B] versus [C]), so that the relative influence of different parameters on spontaneous firing rate was changed. (D) Firing rate sensitivity to D + SA of all optimized models, as a function of their location in the [[Image: see text] , [Image: see text] , [Image: see text] ] subspace. The points shown in (A–C) above are labeled (A, B, and C), with color indicating the magnitude and sign of firing rate sensitivity to D + SA according to the colorscale shown on the right. Arrows show the directions along which sensitivity to D + SA was relatively constant (thin arrow), and along which it changed substantially (thick arrow). Along this principal direction, even the sensitivity sign reversed (yellow circle, marked with “+”). In the 11 models shown as filled circles, firing rate sensitivity was largest to the morphologic parameter D + SA than to any active parameter. Open squares show the four models for which firing rate sensitivity was greater to at least one active parameter than to D + SA.
Figure 5
Figure 5. Systematic Search Across Conductance Space Revealed Continuous Variation in Baseline Spontaneous Firing Rate
Throughout the systematic search, values of [Image: see text] , [Image: see text] , and [Image: see text] ([A], top graph) and [Image: see text] ([A], middle graph, z-axis) remained fixed. In each subpanel, Kp and RCa are held fixed at four different values, labeled A, B, C, and D) ([A], middle graph). A systematic search of the lower 3-D subspace [[Image: see text] , [Image: see text] , [Image: see text] ] for each of these points is shown graphically in A, B, C, and D, respectively. Each model in the [[Image: see text] , [Image: see text] , [Image: see text] ] subspace is shown as a cell whose color represents its spontaneous firing rate, according to the colorscale on the right of subspace A. Uncolored cells were spontaneously silent models. Spontaneous firing rate depended sensitively on a balance between [Image: see text] and [Image: see text] : when [Image: see text] was high but [Image: see text] was low, rates were high, and vice versa. Firing rates also increased with RCa (from A to B and from C to D) and inversely with Kp (from A to C and from B to D).
Figure 6
Figure 6. Sensitivity of Spontaneous Firing Rate Across Parameter Space
(A) Colored symbols show 136 Area II–like candidate models identified during the systematic search of conductance space. Values of [Image: see text] , [Image: see text] , and [Image: see text] (top graph) and [Image: see text] (middle graph, z-axis) were the same for all candidate models; locations along Kp and RCa dimensions (middle graph) and [Image: see text] , [Image: see text] and [Image: see text] dimensions (bottom graph) varied. Some models have been shifted slightly along the [Image: see text] axis to aid in visualization. Red dots indicate candidate models within subspace A (Figure 5A); the blue triangle (“M”) marks the active parameter values used in the systematic search of morphologic space. (B) Sensitivity of spontaneous firing rate to parameter perturbations of the subspace A candidate models as a function of their location in the lower [[Image: see text] , [Image: see text] , [Image: see text] ] subspace. Color indicates sensitivity magnitude and sign according to the colorscale shown at top left. Shown are sensitivities to active parameters RCa and [Image: see text] (top left and right, respectively) and to morphologic parameters D + SA and L + D with constant channel densities (“CD”; bottom left and right). Arrows indicate the principal sensitivity direction across the space.
Figure 7
Figure 7. Sensitivity of Firing Rate Gain Across Conductance Space
(A) Firing rate gain for models within subspace A, according to the colorscale at top right. Uncolored cells did not fire under any of the injected currents. Gain sometimes varied nonmonotonically with [Image: see text] ; compare models labeled 1, 2, and 3. (B) Firing rate versus injected current with fitted gain slopes for Models 1, 2, and 3 shown in (A). Models with low [Image: see text] had high spontaneous firing rates but intermediate gain (Model 3, yellow triangles); gain increased for intermediate [Image: see text] (Model 2, dark red circles), then decreased for high [Image: see text] (Model 1, blue squares). (C) Sensitivity of firing rate gain to active parameters RCa and [Image: see text] (top left and right) and constant density morphologic perturbations D + SA / CD and L + D / CD (bottom left and right) according to the colorscale at top left, for candidate models within subspace A. Arrows indicate the principal direction of global sensitivity trends across the space, which were similar for perturbations of active and morphologic parameters. The sign of sensitivity (marked by “+” and “−”) often changed along this principal direction.
Figure 8
Figure 8. Spontaneous Firing Rate Sensitivity across Constant Density and Constant Numbers Morphologic Spaces
(A) Baseline spontaneous firing rate represented as a colormap across CD morphologic space (colorscale 0–25 Hz; top right). Black dot marks the original morphology used for the conductance space searches. Thick black curve indicates all models matching the original SA. (B) Spontaneous firing rate sensitivities to perturbations of active parameters RCa and [Image: see text] (top row), and morphologic parameters D + SA / CD and L + D / CD (bottom row) of candidate models across the space. Arrows indicate the principal sensitivity trend to each parameter. In the bottom left of (B), filled circles indicate models for which sensitivity to D + SA / CD was greater than sensitivity to all active parameters except [Image: see text] . In the model shown as an open square, sensitivity to two or more active parameters was greater than sensitivity to D + SA / CD. In the bottom right of (B), filled circles and open squares likewise compare the sensitivity to L + D / CD and sensitivities to active parameters. Analogous colormaps across CN morphologic space are shown for (C) baseline spontaneous firing rate and (D) spontaneous firing rate sensitivity to parameter perturbations among candidate models. The black curve in (C) indicates all models matching the original axial resistance (see Materials and Methods). In the bottom row of (D), filled circles show models for which sensitivity to either D + SA / CN or L + D / CN was greater than sensitivities to all active parameters except [Image: see text] and [Image: see text] . In models shown as open squares, sensitivity to three or more active parameters was greater than sensitivity to either D + SA / CN or L + D / CN.
Figure 9
Figure 9. Firing Rate Gain Sensitivity Across Constant Density and Constant Numbers Morphologic Spaces
(A) Colormap representing baseline firing rate gain across CD morphologic space (colorscale 0–400 Hz/nA; top right); the gain of models firing irregularly at 300 or 600 pA was not calculated (gray cells). Black vertical line denotes models with the default value of D; arrow indicates that D was the main driver of baseline gain. (B) Gain sensitivity to RCa and [Image: see text] (top row) and to D + SA / CD and L + D / CD (bottom row). Models shown in gray fired irregularly. Arrows indicate the principal sensitivity direction with increasing D. Sensitivity to [Image: see text] was near zero throughout the space. Analogous colormaps across CN morphologic space are shown for (C) baseline firing rate gain and (D) gain sensitivity to parameter perturbations among candidate models. Arrows in (C) show that decreases in baseline gain followed D; the principal sensitivity directions in (D) follow this trend. Across constant number space, sensitivities to active parameters often had opposite sign for large versus small L values (“−” versus “+” in [D]). Among the candidate models, gain was more sensitive to D + SA than to all active parameters except [Image: see text] and sometimes [Image: see text] .
Figure 10
Figure 10. The Relative Size of Intrinsic Currents Determined the Size of Parameter Sensitivities
(A) The unperturbed somatic voltage trace of the model labeled “B” in Figures 3A and 4D (solid line), compared with the voltage trace after a 5% increase in D + SA (dashed line). Inset barplot shows that spontaneous firing rate sensitivity of this model to [Image: see text] was low (blue), but high to [Image: see text] (red) and D + SA (green). (B) Somatic voltage traces of the model labeled “C” in Figures 3A and 4D. Labeling scheme is analogous to that in (A): the unperturbed model (solid) is compared with a 5% D + SA increase (dashed). Here, firing rate sensitivity was high to [Image: see text] and [Image: see text] but low to D + SA. (C–D) The absolute value of each intrinsic current, normalized by the absolute sum of somatic membrane and axial currents at each time step, for the models shown in (A) and (B), respectively. The Na, K, and K–Ca currents are shown in gray (INa, IK, IK-Ca). Also shown are A-current and NaP current (IA, determined by [Image: see text] , in solid blue; INaP determined by [Image: see text] in solid red) and the somato-dendritic axial current (IAx, determined by morphologic parameters, in heavy solid green). Black arrows indicate artifactual inflection points creased by taking the absolute value, when the unnormalized current reversed sign. For both points B and C, increasing D + SA delayed the next AP and reduced firing rate by increasing IAx; IA and INaP were also perturbed through their interactions with IAx (dashed colored lines). (C) At point B, IAx dominated during the ISI, so that the D + SA perturbation had a large effect on IAx, and INaP. As a result firing rate sensitivity to D + SA was high. (D) At point C, IAx was relatively small throughout most of the ISI. Hence, increasing D + SA had a small effect on the intrinsic currents overall, so that firing rate sensitivity to D + SA was low.
Figure 11
Figure 11. Using Sensitivities To Predict Parameter Compensations for Firing Rate Homeostasis in the Reduced and Realistic Morphologies
(A–B) Parameter perturbations of Model 1 from subspace E (Figure S5, and inset of [E] below). (A) A 15% reduction of D + SA increased firing rate to 14.2 Hz (thick green line) relative to the unperturbed model (12.1 Hz; thin black line); a 14.8% reduction is needed to return to the unperturbed firing rate. (B) As predicted from its firing rate sensitivity (inset barplot), a 10.5% decrease in [Image: see text] compensated almost exactly for the effect of the D + SA perturbation on firing rate in (A) (see Discussion for details). The red trace, shifted up slightly along the vertical axis for visualization, overlays the unperturbed voltage trace very closely. (C–F) Homeostatic parameter compensations can also be predicted in models with realistic morphology. (C) When Model 1 from subspace E was simulated in the realistic AII morphology, a 15% reduction of D + SA increased firing rate by 20.3% relative to the unperturbed value. (D) The compensatory perturbations for Model 1 of two different active parameters ([Image: see text] in red, shifted vertically up; [Image: see text] in blue, shifted down) were predicted from their firing rate sensitivities (inset barplots). The blue and red compensated traces overlie the unperturbed (black) trace very closely, demonstrating excellent compensation. (E–F) Compensatory perturbations predicted from sensitivities computed at a different point in parameter space: Model 3 of subspace E, implemented in the realistic morphology with an analogous 15% reduction of D + SA. Inset in E shows the location of Models 1, 2, and 3 in the [[Image: see text] , [Image: see text] , [Image: see text] ] subspace (also see Figure S5A). For Model 3, firing rate sensitivities to [Image: see text] and [Image: see text] were about twice as large as for Model 1. Accordingly, the predicted magnitudes for Model 3 were about half of those for Model 1; compare (D) and (F). For both models, the close match of red (shifted up), blue (shifted down), and black traces demonstrate that the predicted perturbations of either active parameter compensated almost exactly for the effect of the D + SA reduction ([C] and [E]), even for large predicted increases of [Image: see text] . See Discussion for details.
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References

    1. MacLean J, Zhang Y, Johnson B, Harris-Warrick R. Activity-independent homeostasis in rhythmically active neurons. Neuron. 2003;37:109–120. - PubMed
    1. Prinz AA, Billimoria CP, Marder E. Alternative to hand-tuning conductance-based models: Construction and analysis of databases of model neurons. J Neurophysiol. 2003;90:3998–4015. - PubMed
    1. Prinz AA, Bucher D, Marder E. Similar network activity from disparate circuit parameters. Nat Neurosci. 2004;7:1345–1352. - PubMed
    1. Bucher D, Prinz AA, Marder E. Animal-to-animal variability in motor pattern prediction in adults and during growth. J Neurosci. 2005;25:1611–1619. - PMC - PubMed
    1. Swensen AM, Bean BP. Robustness of burst firing in dissociated purkinje neurons with acute or long-term reductions in sodium conductance. J Neurosci. 2005;25:3509–3520. - PMC - PubMed

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