Qualitative behavior and robustness of dendritic trafficking
QQualitative behavior and robustness of dendritic trafficking
Saeed Aljaberi, Timothy O’Leary, Fulvio Forni
Abstract — The paper studies homeostatic ion channel traf-ficking in neurons. We derive a nonlinear closed-loop model thatcaptures active transport with degradation, channel insertion,average membrane potential activity, and integral control.We study the model via dominance theory and differentialdissipativity to show when steady regulation gives way topathological oscillations. We provide quantitative results on therobustness of the closed loop behavior to static and dynamicuncertainties, which allows us to understand how cell growthinteracts with ion channel regulation.
I. I
NTRODUCTION
Neurons maintain an array of nonlinear conductances thatare mediated by voltage sensitive ion-permeable proteinscalled ion channels. The lifetime of an ion channel is hoursto days, while a neuron typically lives for the lifetime of ananimal [1]. Ion channels are therefore continually replenishedthroughout a neuron. This process operates in closed loop:electrical activity is sensed over long timescales and channeldensities are controlled by negative feedback to homeostati-cally maintain neurons at a reference (average) activity level[2], [3]. Homeostasis is essential for maintaining the electri-cal signaling properties of neurons in the face of biologicalnoise, uncertainty and environmental perturbations, but theunderlying control architecture and constraints are poorlyunderstood [4].Ion channel homeostasis faces significant constraints dueto the complex geometry of neurons [5], [6]. Channel mR-NAs and protein subunits are synthesized at a central locationin the cell body but need to be distributed over an extensivedendritic tree. This is achieved by motor proteins via activeintracellular transport along a network of filaments calledmicrotubules that span the dendritic tree [7], [8], [9], [5],[6]. Although this allows newly synthesized mRNAs andproteins to reach the extremities of a neuron, the traffickingmechanism is thought to be largely blind to the identityof the cargo and the location in the neuron. Cargo istrafficked in bulk throughout the network and selected bylocal sequestration at sites where it is needed. This demand-driven model, known as the sushi belt model [10], can bemodeled mathematically as a compartmental system [9], [5],[11], [6].The closed loop behavior of this system critically dependson the dynamics of the transport system and the geometryof the transport network within the neuron. Furthermore,each step in this process is subject to nonlinearities and
S. Aljaberi is supported by Abu Dhabi National Oil Company(ADNOC). T O’Leary is supported by ERC grant StG 716643FLEXNEURO. S. Aljaberi, T. O’Leary, and F. Forni are with theDepartment of Engineering, University of Cambridge, CB2 1PZ, UK sa798|timothy.oleary|[email protected] uncertainties that are inherent in biological systems. It istherefore challenging to make precise statements about theclosed loop behavior of ion channel regulation using con-ventional system-theoretic tools. Neurons undergo physicalchanges, such as growth, which force them to accommodatechanges to a number of physiological properties. Someproperties, like settling time, need to stay within a certainrange. Other properties, like mRNA synthesis rate, needto increase/decrease depending on the state of the neuron.Indeed, noise and uncertainties are prevalent in biologicalsystems, which requires strong robustness of the feedbackregulation mechanisms.To analyze the closed loop behavior we adopt the novelapproaches of dominance theory and differential dissipativity[12], [13]. We study both regimes of steady regulation andpathological oscillations. We illustrate how key parametersaffect the behavior, in particular how the size of the dendritictree limits the control action, thus the overall regulation per-formance. We provide a detailed robust analysis of the regu-lation mechanisms to parameter uncertainties and unmodeleddynamics. Our analysis is developed in the nonlinear settingand shows that integral control can deal with substantialmodel uncertainties at the cost of reducing performance, asexpected from classical robust control theory [14], [15], [16].In Section II we derive the model and state the biologi-cal assumptions, and show the possible behaviors. SectionIII recalls the main results of dominance theory and p-dissipativity. We provide a classification of possible behav-iors in Section IV. Section V studies the robustness of themodel, to static and dynamic uncertainties. The problem ofgrowth is also discussed. Conclusions follow.II. C
OMPARTMENTAL MODEL OF DENDRITICTRAFFICKING
A. Model formulation and biological assumptions
For expository purposes we consider a neuron as a − compartment system, shown in Figure 1. We show laterhow our analysis is robust to the number of compartments.The first compartment represents the cell body/soma, thesecond and third compartments represent a dendrite, withthe third compartment being the site of cargo usage. Cargoconsists of ion channel precursor ( m i , for ‘mRNA’) which isproduced only in the soma and transformed into functionalion channels ( g i ) in the final compartment. We note thatthere is mixed biological evidence for whether the traffickedprecursor is mRNA or protein. It is possible that mRNAsare trafficked, then channel protein is synthesized locally.Alternatively, channels are synthesized at the cell body then a r X i v : . [ q - b i o . S C ] S e p rafficked. This does not affect our model at our chosen levelof abstraction, so we simply refer to the precursor as mRNA.We model mRNA transport by considering how micro-scopic active transport along microtubules affects concentra-tion m i in each compartment. Inspired by [17], we deriveda mean-field approximation of the random process governedby a Simple Exclusion Principle [18] taking into accountcrowding effects of cargo particles. This results in the follow-ing nonlinear − compartmental model, with finite capacitycompartments, describing dendritic trafficking channel mRNAchannel protein Cell BodynucleusDendrite m m m Fig. 1. Role of dendritic trafficking in neural activity. ˙ m = u ( c − m ) − v f m ( c − m ) + v b m ( c − m ) − wm ˙ m = v f m ( c − m ) − v b m ( c − m )+ v b m ( c − m ) − v f m i ( c − m ) − wm ˙ m = v f m ( c − m ) − v b m ( c − m ) − wm (1) τ g ˙ g = m − g. (2)In (1), m i ∈ [0 , c ] is mRNA concentration in the i th compartment, bounded above by a capacity, c ; v f , v b , w areforward velocity, backward velocity, and mRNA degradation,respectively; u is a control input that regulates mRNAsynthesis. (2) models the dependence of channel protein, g , on mRNA concentration in the third compartment: τ g is a time constant corresponding to protein synthesis. In v f v b ω c m c g ω ω T [ Ca ] target T: somatic [ Ca ] -dependent enzymeg: channel protein m: mRNA [ Ca ] m m m v f v b g Averaging V andcalcium influx Fig. 2. Closed-loop model of regulation of dendritic trafficking. line with experimental data and existing models [19], [20],[3], we assume that channel mRNA synthesis, which occursonly in the first compartment, is dependent on calciumconcentration, [ Ca ] , (Figure 2). Existing models posit thatbiochemical pathways modulate mRNA synthesis accordingto the deviation of calcium concentration from an effectiveset-point, [ Ca ] target . The form of the dynamics of thecontrol signal, T , that transforms the error signal, e =[ Ca ] target − [ Ca ] into, u , in Equation (1) is the subjectof ongoing research. In particular, the question of whether perfect set point tracking is achieved is of great interest. Herewe consider a biologically plausible controller implementingleaky integral control: u = T , τ T ˙ T = e − γT − ϑ ( T ) (3)where γ > sets the degradation rate of T and ϑ ( T ) enforcespositivity and boundedness of T . Here, for simplicity, wetake ϑ ( T ) = a tan (cid:16) πc T (cid:0) T − c T (cid:1)(cid:17) , for < a (cid:28) . In thelimit of γ → , (3) becomes a pure integrator. ϑ ( T ) Tc T Fig. 3. A typical function ϑ ( T ) Calcium concentration varies due to voltage-dependentchannels and may be related to the (quasi-steady state)membrane potential via a saturating monotonic relationship: [ Ca ] = α V/β with parameters α , β that capturecalcium buffering and the voltage sensitivity of calciumchannels [20].Finally, to model the effect of channel protein concen-tration on the membrane potential, V , we consider thestandard single compartment membrane equation, C ˙ V = g leak ( E leak − V ) + g ( E g − V ) , where C is membranecapacitance, g leak is a fixed, leak conductance and the E leak,g terms are equilibrium potentials for each type ofionic conductance. By using a single compartment membraneequation we are assuming that the neuron is equipotential ( V is independent of compartment index). We further assumetimescale separation between the fast voltage fluctuationsand the mRNA synthesis and trafficking mechanisms. Wetherefore set the membrane potential to its quasi-steady state, V := V ss = gE g + g leak E leak g leak + g . B. Model behavior and nominal parameters
TABLE IN
OMINAL P ARAMETER V ALUES v f = 1 v b = 0 . w = 1 g leak = 0 . E leak = − E g = 20 α = 1 a = 0 . β = 1 c T = 10 τ g = 1 τ T = varies [ Ca ] target = 0 . c = 1 γ = 0 n = 3 Using the nominal parameter values in Table I, Figures4(a)-4(c) summarize the behavior of (1),(2),(3) for differentvalues of the integration constant τ T ∈ { , , } . Stableregulation is achieved for large integrator time constant τ T (slow feedback). Performance improves for smaller timeconstants (fast feedback). However, performance rapidlydegrades with the occurrence of pathological oscillationswhen the integral feedback becomes too aggressive ( τ T = 5 ).A nonzero γ in (3) will lead to imperfect tracking. Theanalysis in Section IV shows that these simulations capturethe generic robust behavior of the closed loop system.II. D OMINANCE THEORY IN A NUTSHELL A p -dominant linear system with rate λ ≥ has exactly p slow/dominant modes, whose decay rate is slower than − λ , and n − p fast decaying modes, where n is the systemdimension. The trajectories of the system rapidly convergeto a p -dimensional invariant subspace capturing the steady-state of the system. In state space representation ˙ x = Ax , A ∈ R n × n , linear p -dominance with rate λ is certified bythe Lyapunov inequality A T P + P A + 2 λI < constrainedto symmetric matrices P with inertia ( p, , n − p ) , that is, p negative eigenvalues and n − p positive eigenvalues. p -dominance can be extended to nonlinear systems of the form ˙ x = f ( x ) using the system linearization ˙ δx = ∂f ( x ) δx along arbitrary trajectories [12] ( ∂f ( x ) is the Jacobian of f ). Definition 1:
A nonlinear system ˙ x = f ( x ) is p -dominantwith rate λ ≥ if there exist a symmetric matrix P withinertia ( p, , n − p ) and a positive constant ε such that ∂f ( x ) T P + P ∂f ( x ) + 2 λP ≤ − (cid:15)I (4)for all x ∈ R n . (cid:121) (4) provides a tractable condition for p -dominance throughconvex relaxation, as shown in [12, Section VI.B] and[21, Chapter 4]. It enforces a uniform splitting among theeigenvalues of ∂f ( x ) into p slow eigenvalues to the right of − λ and n − p fast eigenvalues to the left. Our interest inthe property stems from the fact that p -dominance stronglyconstrains the system asymptotic behavior, as clarified by thefollowing proposition from [12, Corollary 1] Proposition 1:
Every bounded trajectory of a p -dominantsystem ˙ x = f ( x ) , x ∈ R n , asymptotically converges to- a unique fixed point if p = 0 ;- a fixed point if p = 1 ;- a simple attractor if p = 2 , that is, a fixed point, a set offixed points and connecting arcs, or a limit cycle. (cid:121) [12, Theorem 2] shows that the asymptotic behavior of a p -dominant system is captured by a p -dimensional dynamics,which thus guarantees simple attractors for p ≤ . Weobserve that a system can be p -dominant and p -dominant, p ≤ p , for different rates λ ≤ λ . In using the theory,wee are typically interested in finding the smallest degreeof dominance, which corresponds to the simplest asymptoticbehavior.Differential dissipativity [12], [13] extends dominance the-ory to open system. We refer the reader to these publicationsfor details. We will use the following notion of p -gain. Definition 2:
An open system ˙ x = f ( x ) + Bu , y = Cx ,with input u , output y , and state x , has p -gain γ with rate λ ≥ if there exist a symmetric matrix P with inertia ( p, , n − p ) and a positive constant ε such that (cid:20) ∂f ( x ) T P + P ∂f ( x ) + 2 λP + C T C − εI P BB T P − γ I (cid:21) ≤ (5)for all x ∈ R n . (cid:121) A straightforward specialization of [12, Theorem 4], seealso [22], provides a differential version of the small gain theorem, which allows us to use the p -gain of a system tocharacterize its robustness in presence of model uncertainties,as in classical robust control theory [14], [15], [16]. Proposition 2:
For i ∈ { , } , let Σ i be systems withinput u i , output y i , and p i -gain γ i with rate λ i = λ ≥ . If γ γ < then the closed loop system given by y = u and y = u is ( p + p ) -dominant. (cid:121) Proposition 2 opens the way to the study of robust attrac-tors that are not fixed points. This is particularly relevant insystem biology. In what follows we will take advantage of thetractability of (4) combined to Proposition 1 to characterizethe steady state behavior of dendritic traffic regulation.Then, we will use the notion of p -gain in combination withProposition 2 to study its robustness.IV. N OMINAL BEHAVIOR AND DIFFERENTIAL ANALYSISOF DENDRITIC TRAFFICKING
Denoting by ˙ x = f ( x ) the closed loop dynamics (1)-(3),Figures 4(d)-4(f) show the position of the eigenvalues of theJacobian ∂f ( x ) for different levels of control aggressiveness,through the selection of values τ T ∈ { , , } and x ,for nominal parameter values in Table I. For readability, weshow only the two right-most eigenvalues of the Jacobian.The others are always to the left of − . . Figures 4(d)-4(f)can be roughly separated in two groups: stable linearization - ∂f ( x ) has stable eigenvalues; Hopf-bifurcation - ∂f ( x ) hasunstable complex eigenvalues. These two groups explain thedifference between stable regulation at steady state and theappearance of oscillations for small τ T . For instance, forlarge τ T = 1000 (slow feedback) there are two real, stableeigenvalues, as shown in Figure 4(d). This is compatiblewith the behavior in Figure 4(a). As the integrator dynamicsbecomes faster, τ T = 80 , the two right most eigenvaluescoalesce and bifurcate, Figure 4(e). Convergence becomesfaster, as shown in Figure 4(b) but damped oscillations mayappear. Finally, for aggressive feedback, τ T = 5 , the complexunstable eigenvalues in Figure 4(f) justify the occurrence ofsustained oscillations in Figure 4(c).The connection between Jacobian eigenvalues and closedloop behavior can be made rigorous through dominanceanalysis, by solving the linear matrix inequality (4) for { m i , g } ∈ [0 . c, . c ] , and using rates λ , λ , λ in Figures4(d)-4(f); CVX [23] was used to numerically solve (4) foreach case of τ T . The first observation is that the systemis always − dominant with rate λ = 0 . . A commonsolution P can be found for τ T ∈ [3 , . This has astriking conclusion: the steady state of the closed loop systemis compatible with planar dynamics, captured by a simpleattractor. This means that for τ T ∈ [3 , the closedloop system either converges to a fixed point or enters intosustained oscillations . This conclusion can be refined: • ( τ T = 1000 ) For slow feedback, λ = 0 . separatesthe two real eigenvalues into two subgroups as shownin Figure 4(d). Feasibility of (4) shows that the systemis -dominant with rate λ = 0 . , which guarantees convergence to a fixed point .
200 400 600 800 1000 t [ C a + ] [Ca ] target (a) τ T = 1000 . t [ C a + ] [Ca ] target (b) τ T = 80 . t [ C a + ] [Ca ] target (c) τ T = 5 .(d) τ T = 1000 . (e) τ T = 80 . (f) τ T = 5 .Fig. 4. (a)-(c) Response of [ Ca ] for different values of τ T where the dendritic trafficking model was simulated with the nominal parameter valuesin Table I and Initial condition x = [0 .
2; 0 .
2; 0 .
2; 0 .
2; 0 . T .(d)-(f) two right-most eigenvalues of the Jacobian of the closed loop given by (1)-(3). ForFigure (e) the right-most eigenvalue does not cross the imaginary axis. Spectra were obtained by sampling . ≤ x i ≤ . , where the black and blue dotsdepict the movement of the two right-most eigenvalues. . TABLE IIS
OLUTIONS TO (4)
FOR UNCERTAINTIES IN T ABLE
III. − dominance − dominance P = . . . − . − . . . . − . − . . . . − . − . − . − . − . . . − . − . − . . − . P = . − . − . − . − . − . − . − . − . − . − . − . − . − . . − . − . − . . . − . − . . . − . • ( τ T = 80 ) As the integrator dynamics become faster thetwo right most eigenvalues bifurcate and − dominanceis lost (Figure 4(e)). However, the system is still − dominant locally with rate λ = 0 . This guarantees local convergence to the fixed point . • ( τ T = 5 ) For aggressive integrator dynamics the systemis − dominant with rate λ = 0 . . It cannot be -dominant (complex right-most eigenvalues) and it can-not be -dominant (unstable eigenvalues). -dominancecombined with the instability of the fixed point guar-antees that sustained oscillations are the only possiblesteady state behavior .V. R OBUSTNESS AND GROWTH
A. Parametric uncertainties
The analysis above shows how the feedback time constant τ T affects regulation. We now study robustness to otherphysiologically relevant parameters (such as velocities andlength) using dominance theory, looking at the three regimes τ T ∈ { , , } . For τ T = 1000 a stable closed loopbehavior is preserved for any uncertainty in Table III (leftcolumn). For τ T = 5 , the robustness of the oscillatory regimeis guaranteed for parameter ranges specified in Table III(right column). A local robust analysis is also developed for τ T = 80 . This is a fragile case for dominance analysis, which we address numerically by looking at specific local regions. TABLE III − dom: τ T =1000 , λ =0 .
05 2 − dom: τ T =5 , λ =0 . v f [0 . , .
2] [0 . , . v b [0 . , .
1] [0 . , . τ g [0 . , .
4] [0 . , . For τ T = 1000 , the controller guarantees robust -dominance with rate λ = 0 . to uncertainties in TableIII (left column). Indeed, the matrix P in Table II is asolution to (4) for all parametric uncertainties in Table III(left column). Robust stable regulation is thus guaranteed forthese uncertainties. Stable regulation is also preserved whenthe velocity constants are replaced by nonlinear functions v f ( m i − , m i , m i +1 ) and v b ( m i − , m i , m i +1 ) whose slopes v (cid:48) f and v (cid:48) b belong to the intervals defined in Table III.For τ T = 5 , the closed loop system is robustly -dominantwith rate λ = 0 . to uncertainties in Table III (right column).This is certified by the matrix P in Table II which is asolution to (4) for all parametric uncertainties in Table III(right column). As discussed in other sections, − dominanceis not sufficient to claim robust oscillations. However, theunique equilibrium of the system is always unstable forparameters in Table III (right column) which, combined with -dominance, guarantees robust oscillations.For τ T = 80 , the closed loop is moving from a stable toan oscillatory regime (complex stable poles in the Jacobian).High sensitivity to parameter variations is thus expected.Table IV shows the trade-off between parameter ranges andsize of the region of − dominance. TABLE IV − DOMINANCE : τ T = 80 , λ = 0 . around x ∗ around x ∗ around x ∗ v f [0 . , .
2] [0 . , .
45] [0 . , . v b [0 . , .
1] [0 . , .
3] [0 . , . τ g [0 . , .
2] [0 . , .
45] [0 . , . B. Growth
How does a neuron tune its transcription rate in the pres-ence of growth? A bigger neuron requires more biomoleculesto be synthesized and their traveling distance is longer. Withthese variations, can a neuron withstand and maintain a stablenominal behavior? Growth can be modeled in two ways:by increasing the number of compartments or by adaptingcapacity and velocity parameters. We adopt the latter forsimplicity.We consider -dimensional growth, where L represents theneuron’s total length. The identity c = L/n relates length L to compartment’s capacity c and to compartments number n . Growth corresponds to larger L thus larger capacities.Forward and backward speeds are also updated accordingly.Starting from the microscopic picture, suppose that eachcompartment can fit c number of molecules as shown inFigure 5. The figure shows a large compartment z j ofcapacitance c and its constituent unit compartments x i ’s,each of capacity . The rate of change of molecules incompartment z j is given by ˙ z j = vx i − (1 − x i ) − vx i + c − (1 − x i + c ) (6)where the internal exchange of molecules sum to zero.We focus on particles that enter and leave z j , assumingthat particles are homogeneously distributed and spatiallyindistinguishable (well-mixed) in each compartment z j , thatis, x i = x i +1 · · · = x i + c − = z j c . (7)Substituting (7) into (6), we get ˙ z j = v z j − c (cid:16) − z j c (cid:17) − v z j c (cid:16) − z j +1 c (cid:17) = vc z j − ( c − z j ) − vc z j ( c − z j +1 ) . (8)Equation (8) shows that increasing L the compartment sizeincreases linearly and the velocities scale with /c orequivalently /L . In summary, growth is modeled by thefollowing parameter scaling in (1): v f → v f c , v b → v b c , c = Ln . (9)Within this modeling framework, the question of growthreduces to a question of robustness to parameter variations.The first question is: given an integrator time constant τ T , x i x i +1 x i + c − v v v v v x i + c x i − vv z j z j +1 z j − Fig. 5. A schematic representation of equations (6)-(7). how much can the neuron grow before loosing stability?We answer through − dominance, by deriving intervals oflength L that guarantee -dominance for a fixed time constant τ T , as shown in Figure 6(a). As expected, stable regulationfor longer neurons requires less aggressive feedback (larger τ T ). For any time constant τ T , there is a threshold lengthafter which -dominance is lost. This regime is characterizedby the emergence of damped oscillations, which eventuallydegrade into sustained oscillations for longer lengths. Infact, Figure 6(b) shows that -dominance of the closed loopis preserved for large variations (both on L or τ T ) withlimit cycles appearing when the time constant is sufficientlysmall or the length is sufficiently large, that is, when theequilibrium of the system loses stability. L T loss of 1-dominance1-dominance (a) − dominance with λ = 0 . . L T loss of 2-dominance2-dominance(unstable)2-dominance(stable) (b) − dominance with λ = 0 . .Fig. 6. Trade-off between τ T and L for − and − dominance. C. Unmodeled dynamics
Both growth and parametric variations have been modeledin previous sections as static uncertainties. We now consider dynamic uncertainties typically arising from unmodeled dy-namics and modelling simplifications.We will model theseuncertainties as (possibly nonlinear) -dominant dynamicperturbations, ∆ and ∆ , acting on the nominal closedloop as shown in Figure 7. ∆ corresponds to additiveperturbations, such as neglected transport phenomena. ∆ corresponds to multiplicative uncertainties such as neglectedfast dynamics in protein synthesis.We assess the robustness of the closed loop using thenotion of p -gain in Section III and the small gain intercon-nection in Proposition 2, which guarantees that perturbationsdo not affect the the dominance of the closed loop if theproduct of the nominal gain and of the perturbation gain isless than one. Indeed, for the nominal parameters in Table I,solving (5), the nominal closed loop in Figure 7 has -gain γ cl = 0 . from u to y and and -gain γ cl = 2 . from u to y , both with rate λ = 0 . . -dominance of theclosed loop, i.e. steady regulation, is thus preserved for anyperturbation ∆ whose -gain γ satisfies γ < /γ cl withrate λ . -dominance is also preserved when ∆ has -gain γ < /γ cl with rate λ . Ca ] target + T m ∆ ∆ gϕ ( . ) − e [ Ca ] y u y u + + Fig. 7. A schematic showing how the unmodeled dynamics affect thenominal closed-loop as dynamic perturbations.
20 40 60 80 100 N Fig. 8. -gain γ of ∆ = Σ N − Σ for ≤ N ≤ and rate . . As an example we study closed loop regulation when the -compartmental model of transport Σ is replaced by amore detailed N -compartmental model Σ N , N > . Forthis case ∆ represents the mismatch dynamics Σ N − Σ .For simplicity we restrict our analysis to linear transportmodels, that is, we take Σ as in (1) but ignore compartmentsaturation. Σ N is also a linear compartmental system. Itsparameters are scaled according to Section V-B, but thistime keeping a constant length L and varying the numberof compartments. Figure 8 shows how the -gain γ (rate . ) of ∆ changes with the number of compartments. Forthe nominal parameters in Table I, γ peaks at . , whichguarantees that the closed loop behavior remains unchangedif we replace our -compartmental transport model with amore detailed transport model based on ≤ N ≤ compartments.A similar analysis can be developed to account for un-modeled protein dynamics to show that sufficiently fastreactions can be safely neglected. These examples show theflexibility of the framework in systems biology for capturingheterogeneous families of perturbations, mimicking classicalrobust control. We note that the approach is not limited tolinear perturbations and can be extended beyond fixed pointanalysis to study robust oscillations via -dominance.VI. C ONCLUSION
We presented a nonlinear model of dendritic traffickingthat captures spatial and crowding effects. We studied in-tegral regulation of dendritic trafficking in the context ofion channel regulation, providing results on its robustness toparametric and dynamic uncertainties, and to growth. A largenumber of questions remain unanswered. Our analysis ofgrowth, for example, is based on the variation of the dendritelength but neurons develop by branching, expanding theirdendritic trees within complex morphologies and with vary-ing diameters among different sections. This is an importantresearch direction that we believe can also be addressed withthe tools adopted in this paper. Another promising directionis the study of regulation of synapses, or non-homogeneouscompartments with different conductance levels. Finally, in this study we assumed that the soma is responsible for theneuron’s entire regulation process. However, experimentalstudies suggest that regulation is achieved by coordinationbetween global (integral control) and local (degradation)mechanisms. This is an intriguing direction that we willexplore in future publications.R
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