Dendritic trafficking: synaptic scaling and structural plasticity
DDendritic trafficking: synaptic scaling and structural plasticity
Saeed Aljaberi , Timothy O’Leary , Fulvio Forni Abstract — Neuronal circuits internally regulate electrical sig-naling via a host of homeostatic mechanisms. Two prominentmechanisms, synaptic scaling and structural plasticity, arebelieved to maintain average activity within an operating rangeby modifying the strength and spatial extent of network con-nectivity using negative feedback. However, both mechanismsoperate on relatively slow timescales and thus face fundamentallimits due to delays. We show that these mechanisms fulfillcomplementary roles in maintaining stability in a large network.In particular, even relatively, slow growth dynamics improvesperformance significantly beyond synaptic scaling alone.
I. INTRODUCTIONNeurons are excitable cells that can interconnect withthousands of other cells. The excitable properties of neuronsas well as the synaptic connections between them dependon ion-permeable channels and receptor proteins that haverelatively short half-lives and thus require continual replen-ishment [1]. Furthermore, learning and memory in biologicalneuronal networks is implemented by adaptation of thedensities of these signalling molecules at synapses as wellas formation and elimination of synaptic connections. Suchadaptation in connectivity can potentially lead to destabi-lization of activity in large networks. Homeostatic synapticscaling, a form of synaptic plasticity, normalizes inputs tokeep neuronal activity within an operating range [2].However, the complex geometry of neurons impose sig-nificant constraints on synaptic scaling and ion channelhomeostasis. Channel mRNAs and protein subunits need tobe actively transported via protein motors on cytoskeletalcomponents called microtubules [3], [4], [5]. The synthesisprocess of ion channels and receptor proteins depends on afeedback signals, including calcium influx [6], [7]. Previouswork showed that transport processes suffer from severedelays [8], and can pose a potential source of instability inthe presence of cellular feedback control [9].Activity-dependent structural plasticity is another formhomeostatic mechanism. Unlike synaptic scaling, structuralplasticity encompasses morphological changes, such as out-growth/shrinking of dendrites or axons, dendritic branching,and formations/elimination of synapses [10]. Such changesin the neuron’s structural properties happen in response tovariations in the electrical activity [11], similar to synapticscaling. These changes are particularly prevalent duringdevelopment as neurites grow and as connections first form. *S. Aljaberi is supported by Abu Dhabi National Oil Company (AD-NOC). T. O’Leary is supported by ERC grant StG 716643 FLEXNEURO. The authors are with the Department of Engineering,University of Cambridge, Cambridge CB2 1PZ, U.K. sa798|timothy.oleary|[email protected]
Thus, alterations to this process can profoundly and perma-nently affect the function of a mature network [12], [13].Both synaptic scaling and structural plasticity are crucialhomeostatic processes, but the contributions of each to thelong term dynamics and stability of neuronal circuits remainspoorly understood. We address this by formulating a systemtheoretic model that captures both processes from first prin-ciples. We derive conditions under which these mechanismsguarantee stability. We derive a necessary condition forhomeostasis, that is, stability of a network, that relatesbiosynthesis rates of signalling proteins to morphologicalparameters of neurons (typical neurite length). Finally, weshow that structural plasticity provides a stabilizing influencewhen synaptic scaling is aggressive.In Section II and III we model synaptic scaling growthadaptation, also stating state the relevant biological assump-tions. Stability and general properties of the closed loopare illustrated and discussed in Section IV Conclusions andfuture research directions follow. The proof of the maintheorem is in appendix. mRNAprotein
Dendrite Cell Bodynucleus m L m m n g g n g n − g n − u Fig. 1. Cargo trafficking and ion-protein synthesis.
II. S
YNAPTIC SCALING
Ion channels and receptors are responsible for the electri-cal activity of the neuron and are distributed throughout thecell. These signalling components are continually replenishedthrough synthesis and transport as intracellular cargo. Acommon kind of cargo is mRNA that is used to synthesizeion channel proteins locally in dendrites. The concentrationsof these cargo is regulated by feedback mechanisms, typi-cally based on the average neural activity which correlateswith calcium concentration. In what follows we provide abasic model of these three main components, namely cargotrafficking, ion-protein synthesis, and feedback regulation . It is also possible that the ion-channel proteins are synthesized directlyin the cell-body then transported to the synapses, where they are inserted.This would not introduce fundamental modifications to our model. a r X i v : . [ q - b i o . N C ] N ov . Cargo trafficking We model cargo trafficking as a nonlinear active transportphenomenon . Unlike diffusion, which depends on concen-tration gradients, active transport requires energy and isperformed by protein motors, typically dyenin, kinesin, ormyosin. These motors interact with microtubules and actinfilaments, which take the role of track rails. Crowding andfinite size effects must also be taken into account, typi-cally through classical statistical physics modeling [14] andmean-field approximation, leading to simple nonlinear ODEsmodeling saturation effects. In this paper the complex cargodynamics is approximated by the following compartmentalmodel: ˙ m = u − m ( c − m ) − ω m m ˙ m = m ( c − m ) + v b c ( c − m ) m − v f c ( c − m ) m − ω m m ˙ m i = − v f c ( c − m i +1 ) m i + v b c ( c − m i ) m i +1 + v f c ( c − m i ) m i − − v b c ( c − m i − ) m i − ω m m i ˙ m n = v f c ( c − m n ) m n − − v b c ( c − m n − ) m n − ω m m n . (1)(1) is a basic nonlinear transport dynamics. From Figure1, m represents the mRNA concentration in the soma, m i ∈ [0 , c ] represents mRNA concentration in dendriticcompartment i ≥ , v f and v b are the forward and backwardtransport rates, respectively, ω m is the mRNA degradationrate, u represents mRNA production, and c represents thefinite capacity of a single compartment, to model crowding.We assume crowding effects do not occur in the soma, sinceit is significantly larger than dendritic compartments.Notably, the transport rates v f ( c − m i ) and v b ( c − m i ) are scaled by the square of the capacity, c . The purpose isto adapt transport rates in accordance with the compartmentsize. In contrast to increasing the number of compartments,this enables modeling of growth by adaptation of capacityand transport rates. For instance, consider forward transportwith normalized capacity c = 1 . A large compartment withcargo denoted by z j can be modeled as a collection of smallercompartments whose cargo is denoted by y i , as shown inFigure 2. From (1) we get the generic transport dynamics ˙ z j = vy j − (1 − y i ) − vy i + c − (1 − y i + c ) , where the internal exchange of molecules sums to zero.Focusing on the molecules that enter and leave z j , andassuming that the molecules are homogeneously distributedthroughout the compartment z j (well-mixed), we can write y i = y i +1 = · · · = y i + c − = z j c . Thus, substituting the latter in the equation of ˙ z j , 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 ) . which justifies the model in (1). y i y i +1 y i + c − v v v v v y i + c y i − vv z j z j +1 z j − Fig. 2. Microscopic picture of transport.
B. Ion channel synthesis and electrical activity
The synthesis of ion channels g i from spatially distributedmRNA concentrations m i is a complex biochemical processthat we approximate as a first order process ˙ g i = sm i − ω g g i , (2)where s is the translation rate and ω g is the degradation ofion channel proteins. The aim of (2) is to capture roughly thetemporal features and the static gain of this complex process.In first approximation, the effect of ion-channels g i on aneuron’s electrical activity is captured by a simple leaky-integrator model, based on a single-compartment, Ohmiccurrent balance relationship: C ˙ V = g leak ( E leak − V ) + g avg ( E g − V ) . V is the membrane potential, C is themembrane capacitance, g leak is a fixed leak conductance, E leak , E g are equilibrium potentials, and g avg = 1 n n (cid:88) i =1 g i (3)is the averaged sum of ion-channel conductances. Sincevoltage fluctuations occur on a time scale that is faster thancargo trafficking m i and protein synthesis g i , we resolve themembrane voltage V to a quasi steady state V = g avg E g + g leak E leak g leak + g avg . Finally, following [15], we observe that the electrical activityof the neuron affects calcium concentration, captured by asigmoidal monotone relation. For simplicity, we take [ Ca +2 ] = α − V /β ) where α and β shape the sensitivity of [ Ca +2 ] to V . In whatfollows we will use the non-increasing function h [ Ca +2 ] = h ( g avg ) (4)to denote the composition of voltage V and calcium [ Ca +2 ] equations, as illustrated in Figure 3. C. Ion channel / cargo regulation
Homeostatic control of neural activity is achieved byregulation of intracellular calcium [ Ca +2 ] to a certain target [ Ca +2 ] target . This is achieved by controlling the productionof mRNA in the soma in agreement to a (leaky) integralcontrol law ˙ u = k u e − ω u u (5)here e = [ Ca +2 ] target − [ Ca +2 ] , k u is the integral gain,and ω u is the degradation rate, typically small. As for theprotein synthesis in the previous section, feedback control isachieved through a complex bio-chemical process. A detailedmodel of this process is beyond the scope of this paper. Theinterested reader is referred to [15] and references therein.(5) provides a basic approximation describing time-scale andsteady-state gain of this complex process.Feasibility of the steady state [ Ca ] = [ Ca ] target for the closed loop (1)-(5) is guaranteed by the followingassumption. Assumption 1:
The parameters of (1)-(5) satisfy E g + β ln α [ Ca +2 ] target (cid:54) = 0 and g leak (cid:54) = 0 . The closed loop given by Equations (1)-(5) describes theso-called phenomenon of synaptic scaling, a global home-ostatic mechanism that prevents activity from building upand eventually reaching pathological levels. The neuron, viafeedback, reduces/increases the cargo density in the system,which in turns reduces/increases the ion-channel density,ultimately normalizing the electrical activity of the neuron.For a detailed study of this feedback mechanism the readeris referred to [2].III. S
TRUCTURAL PLASTICITY AS GROWTH DYNAMICS
Structural plasticity involves multiple morphologicalchanges that happen in response to perturbations in elec-trphysiological activity [10]. Morphological changes in-clude dendritic/axonal length variations (during develop-ment), synapse formation and elimination (dendritic spinesand axonal boutons), and branching. In our model, suchvariations are captured by a single length parameter L ,describing the average length of a dendritic arbor.We model structural plasticity as a growth process that isdirectly coupled to the neurons’s average activity, capturedby calcium concentration [ Ca +2 ] . The basic idea is that shortdendrites have fewer connections, thus a reduced electricalactivity. Likewise, long dendrites potentially make moreconnections, thus enjoy stronger electrical activity. In thissense, L can also be considered as an abstract indicator ofthe connectivity of the neuron. Then, a feedback mechanismadjusts L to achieve homeostasis in a way that is not atall dissimilar from synaptic scaling [16], [17], [18]: aboveset-point ( e < ), L must shrink to reduce the number ofsynapses (pruning) / weakening existing connections, thusreducing the overall electrical activity; conversely, below setpoint ( e > ), L must increase for the neuron to reach out toother neurons / strengthening existing connections, ultimatelyincreasing the level of electrical activity.Based on these physiological observations, we model thegrowth dynamics using the nonlinear first order process τ ˙ L = φ ( e ) − ω L L , c = Ln (6)where τ (cid:29) is a slow time constant reflecting the slow dy- namics of growth , and ω L is the degradation or disassemblyrate of the molecules that are responsible for synthesis of thenew dendritic components, such as tubulin. The function φ , φ (0) = 0 , is a monotonically increasing function in the error e (see Figure 3). g avg e e (e=0) = 0 Fig. 3.
Left: shape e = [ Ca +2 ] target − h ( g avf ) , which incorporatesthe static map h in (4) (with β = 5 ). Right: an example of the monotoneincreasing function φ . The graph corresponds to (7). If we interpret L as a connectivity indicator, the slope φ (cid:48) captures the density of the surrounding neurons or richnessof the network; a steeper φ means there are more potentialconnections to be made or removed, for the same amountof growth. We emphasize that growth in (6) eventuallycorresponds to a simple variation of capacity, which alsoaffects the forward and backward transport rates in (1)-(5)Both synaptic scaling (1)-(5) and growth dynamics (6)aim to achieve the same objective: regulating the neuron’saverage activity around an (approximate) set-point. Thedifference is that synaptic scaling occur at a fast timescaleand it works by modulating (globally) the number of ionchannels g i in the system. In contrast, growth occurs at aslow timescale and it changes compartments’ capacities tomodulate the maximum allowable g i in each synapse. Figure4 provides an illustration of the complete closed loop (1)-(6). m gV [ Ca +2 ] [ Ca +2 ] target u − + e Lφ ( . ) fast system slow system Fig. 4. Complete closed loop (1)-(6) block diagram.
IV. H
OMEOSTASIS BY FAST SYNAPTIC SCALING ANDSLOW GROWTH ADAPTATION
For any fixed dendritic length L , synaptic scaling (1)-(5)is a stable process if the feedback gain k u is sufficiently In biological neurons, growth rates are on the order of days or weeks[19], [20], while active motor-assisted transport is of the order of hours [8].For example, in
C. elegans , it was found that they grow at an average rateof . µm/s , while active transport rates are O (1 µm/s ) [21], [22]. mall. In fact, the aggressiveness of the control action isfundamentally limited by the presence of transport, typicallyintroducing a phase delay that limits the feedback gain.Growth adaptation is also a stable process, naturally occur-ring at a slower time scale than synaptic scaling. Thus, bytime-scale separation, the combination of synaptic scalingand growth adaptation leads to a stable closed loop (1)-(6),for a sufficiently slow growth time constant τ . Theorem 4.1:
Under Assumption 1, there exists a maxi-mal feedback gain ¯ k u > and a minimal time constant ¯ τ > such that, for every ≤ k u < ¯ k u , τ > ¯ τ , and [ Ca ] target the closed-loop (1)-(6) has a globally exponentially stableequilibrium.Taking advantage of the theoretical result in Theorem 4.1,we study the system’s response under different biologicallyrelevant situations, to better understand the interplay betweensynaptic scaling and growth dynamics. With this aim, we set φ ( e ) in (6) as in [16] φ ( e ) = 1 −
21 + exp( e/η ) , (7)and we simulate the system for the parameters in Table I. v f = 1 v b = 0 . ω m = 0 . n = 2 E leak = − E g = 20 ω L = 0 . η = 0 . β = 1 α = 1 ω g = 0 . τ = 10 [ Ca ] target = 0 . g leak = 0 . ω u = 10 − TABLE I
The first observation is that growth adaptation guaranteeshomeostasis even if synaptic scaling is insufficient . First ofall, note that the averaged sum of ion-channel proteins islimited by g avg = 1 n n (cid:88) i =1 g i = snω g n (cid:88) i =1 m i ≤ snω g nc = sω g c . Thus, regulation is feasible (cid:39) e = [ Ca +2 ] target − [ Ca +2 ] = [ Ca +2 ] target − h ( g avg ) , only if the desired steady state satisfies [ Ca +2 ] target ≤ h (cid:18) sω g c (cid:19) . (8)Inequality (8) fundamentally relates the calcium target / thedesired level neural activity to the morphological parameter c = L/n . It shows that, without growth adaptation, highlevels of neural activity ( [ Ca +2 ] target large) cannot be at-tained in closed loop because of the finite capacity of cargotransport. However, taking advantage of growth adaptation,the neuron can develop its morphology to reach the desiredcalcium target.These two cases are illustrated through simulation, withina comparison between synaptic scaling without growth adap-tation (1)-(5), and synaptic scaling with growth adaptation(1)-(6). Results are summarized in Figure 5. Left and rightgraphs shows the calcium [ Ca +2 ] trajectory and the length L trajectory, respectively. Dashed lines correspond to synaptic scaling without adaptation, while continuous lines corre-spond to the growth adaptation case. Figure 5 shows thecase in which [ Ca +2 ] target = 0 . is not compatible with thethe initial capacity c = L/n = 0 . /n in the sense of (8).The dashed line shows that synaptic scaling without growthadaptation is stable but far from target. This is not the caseof synaptic scaling with growth adaptation, whose calciumtrajectory asymptotically converges to [ Ca +2 ] target , takingadvantage of the increased average length L , thus of largercapacity c .Considering L as a connectivity parameter, the biologicalinterpretation is that the neuron is below its target activitylevel and therefore attempts to increase its activity by ex-tending its dendritic tree to form new connections. Likewise,considering L as a morphological parameter, the neuronincreases the size of its spines to allow more ion channelsto flow to the synapse, which also increases the electricalactivity. t [ C a ] + [Ca +2 ] target t L Fig. 5. Average activity and length for the synaptic scaling model (1)-(5)(dashed) and synaptic scaling model with growth dynamics (1)-(6) (solid). k u = 0 . and L = 0 . . The second observation, derived from simulations, is that growth adaptation may compensate for pathological oscilla-tions, enabling more aggressive synaptic scaling . Aggressivefeedback gains k u may lead to pathological oscillations insynaptic scaling [9], as shown in Figure 6(a). However,these oscillations are dampened through growth adaptation,as shown in Figure 6(b), reaching the desired set-point. Theintuition is that (6) is essentially a low pass filter thereforeit filters calcium oscillations, extracting the oscillations bias.The overall growth adaptation is thus driven by this bias.When the bias is above the desired calcium target, as inFigure 6, the average length will reduce, stabilizing the os-cillations. The biological interpretation is that large neuronsreduces their size when their average electrical activity isirregular (oscillatory).The last observation is that inadequate timescale sep-aration leads to fragility . Theorem 4.1 guarantees closedloop stability under the strong hypothesis of time scaleseparation between synaptic scaling and growth adaptation.The simulations in Figure 7 shows that time scale separationis actually needed for stability. As τ decreases the systemstability becomes more fragile. Reducing τ produces dampedoscillations and a further reduction eventually leads to sus-tained oscillations, for smaller values of τ .
500 1000 1500 t [ C a ] + t L (a) Synaptic scaling model (1)-(5) t [ C a ] + [Ca +2 ] target t L (b) Synaptic scaling model with growth dynamics (1)-(6)Fig. 6. grow adaptation (6) increases the maximum allowable ¯ k u .Simulations were done with k u = 0 . and L = 0 . . For readability,the calcium trajectory in the left graph of Figure 6(a) is represented on thereduced domain ≤ t ≤ . V. CONCLUSIONS
We presented a model of nonlinear dendritic traffickingwith growth adaptation to study two distinct homeostaticmechanisms: synaptic scaling and structural plasticity. Westudied the interplay between the two and how timescaleseparation provides the means to improve the overall perfor-mance of the system. Using contraction arguments combinedwith singular perturbation theory, we proved exponentialstability of the closed-loop equilibrium and we discussedseveral features of the closed-loop system, supported by sim-ulations. Our growth model is very simple. Future researchwill focus on extended modeling of growth.R
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ROOF OF THEOREM ≤ k u < ¯ k u , provided that ¯ k u is sufficiently small.Thus, stability of the closed loop (1)-(6) follows from [23,Theorem 11.4], under the assumption of sufficient time-scaleseparation τ (cid:29) . Part 1: contraction / stability of the fast system
First we multiply equations (1)-(6) by (cid:15) = τ . We startby proving the stability of the fast system (1)-(5). The timederivative in the equations below refers to the scaled time ˜ t = tτ . In the fast timescale, the slow variable L is consideredas constant. The linearized dynamics of the time-scaled fastsystem (1)-(5) reads (cid:15)δ ˙ m = ∂f∂m δm + Bδu (9) (cid:15)δ ˙ g = Sδm − Ω g δg(cid:15)δ ˙ u = − k u ∂h∂g (cid:18) T gn (cid:19) T n δg − ω u δu. where f ( m, L ) is the right-hand side of (1), S = diag { s } , Ω g = diag { ω g } , m = [ m , . . . , m n ] T , g = [ g , . . . , g n ] T and g avg = n (cid:80) ni g i = T gn , where is a vector of ones.We need to show that (9) is a contracting system, whichimplies the existence of a globally exponentially stableequilibrium when the contracting distance is a norm. We firstnote that ∂f∂m T + ∂f∂m ≤ − ω m I < . Take the differentialLyapunov function V = ρ m δm T δm + ρ g δg T δg + δu T δu . The coefficients ρ m > and ρ g > will be defined later.Its time derivative reads ˙ V = ˙ V m + ˙ V g + ˙ V u (10)where ˙ V m = ρ m (cid:32)(cid:20) ∂f∂m δm + Bδu (cid:21) T δm + δm T (cid:20) ∂f∂m δm + Bδu (cid:21)(cid:33) < − ρ m ω m δm T δm + ρ m Bδu T δm , ˙ V g = ρ g Sδm T δg − ρ g Ω g δg T δg, and ˙ V u = − k u ∂h∂g (cid:18) T gn (cid:19) T n δgδu − ω u δu T δu. Therefore, (10) satisfies ˙ V < − ρ m ω m δm T δm + ρ m Bδu T δm + ρ g Sδm T δg (11) − ρ g Ω g δg T δg − k u ∂h∂g (cid:18) T gn (cid:19) T n δgδu − ω u δu T δu< − ρ m | ω m || δm | + ρ m | B || δu || δm | + ρ g | S || δm || δg |− ρ g λ min (Ω g ) | δg | + k u (cid:12)(cid:12)(cid:12)(cid:12) ∂h∂g (cid:18) T gn (cid:19) T n (cid:12)(cid:12)(cid:12)(cid:12) | δg || δu |− ω u | δu | . The right-hand side of (11) is bounded by | δm || δg || δu | T − ρ m | ω m | ρ g | S | ρ m | B | ρ g | S | − ρ g λ min (Ω g ) k u (cid:12)(cid:12)(cid:12) ∂h∂g (cid:16) T gn (cid:17) T n (cid:12)(cid:12)(cid:12) ρ m | B | k u (cid:12)(cid:12)(cid:12) ∂h∂g (cid:16) T gn (cid:17) T n (cid:12)(cid:12)(cid:12) − ω u (cid:124) (cid:123)(cid:122) (cid:125) − Q | δm || δg || δu | . Next we show that
Q > , using the Sylvester criterion.This guarantees contraction, therefore global exponentialstability of the fast system equilibrium We start by findingconditions under which the leading principal minors of ρ m | ω m | − ρ g | S | − ρ m | B |− ρ g | S | ρ g λ min (Ω g ) − k u (cid:12)(cid:12)(cid:12) ∂h∂g (cid:16) T gn (cid:17) T n (cid:12)(cid:12)(cid:12) − ρ m | B | − k u (cid:12)(cid:12)(cid:12) ∂h∂g (cid:16) T gn (cid:17) T n (cid:12)(cid:12)(cid:12) ω u (12)are positive. We will use the following facts: | ω m | = ω m , | S | = s , λ min (Ω g ) = ω g , | B | = 1 .The first principal minor must satisfy ρ m ω m > , whichis true. The second principal minor must satisfy ρ m ρ g ω m ω g − ρ g s > . (13)The last principal minor must satisfy ρ m ρ g ω g ω g ω u − ρ g ρ m sk u (cid:12)(cid:12)(cid:12)(cid:12) ∂h∂g (cid:18) T gn (cid:19) T n (cid:12)(cid:12)(cid:12)(cid:12) + − ρ m ρ g ω g − ρ m ω m k u (cid:12)(cid:12)(cid:12)(cid:12) ∂h∂g (cid:18) T gn (cid:19) T n (cid:12)(cid:12)(cid:12)(cid:12) − ρ g s ω u > which can be re-arranged as ρ m ρ g ω g ω g ω u − ρ g s ω u − ρ m ρ g ω g > k u ρ m (cid:12)(cid:12)(cid:12)(cid:12) ∂h∂g (cid:18) T gn (cid:19) T n (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) ρ g s + ω m k u (cid:12)(cid:12)(cid:12)(cid:12) ∂h∂g (cid:18) T gn (cid:19) T n (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) (14)n order for the above inequality to hold, we need the left-hand side to be positive and larger than the right hand side.So, (13) and (14) hold if we select1) ρ g < ρ m ω m ω g s .2) ρ m < ω u ω m .3) ≤ k u < ¯ k u is sufficiently small.Under these conditions, ˙ V ≤ − ¯ λV for some ¯ λ > .The exponential decay of the differential Lyapunov func-tion guarantees global incremental exponential stability ofthe fast system, [24, Theorem 1]. This implies global expo-nential stability of the equilibrium of the fast system. Part 2: contraction / stability of the reduced system
We study the stability of the reduced system given by (6)for e computed from the fast system at steady state. Thus,as a first step, we study the monotonicity properties of thestatic relationship between e and L , denoted by e = r ( L ) .Define M := n (cid:80) i =1 m i and G := n (cid:80) i =1 g i . At steady state, ˙ m = 0 , ˙ M = 0 , ˙ G = 0 , ˙ u = 0 , we have u − m (cid:18) Ln − pM (cid:19) − ω m m m (cid:18) Ln − pM (cid:19) − ω m M sM − ω g G k u e − ω u u , (15)where we have written m at steady state as m = pM with < p < . For simplicity, we use x to denote the vector x = [ m ; M ; G ; u ] T , and R ( x, L ) to denote the right-handside of (15).The monotonicity of the static relationship e = r ( L ) canbe determined from the equation R ( x, L ) = 0 . For instance, ∂R∂x δx + ∂R∂L δL = 0 , which gives δx = − (cid:20) ∂R∂x (cid:21) − ∂R∂L δL . (16)We observe that the inverse (cid:2) ∂R∂x (cid:3) − must exists sincethe fast system is contractive. Furthermore, the error e =[ Ca +2 ] target − h ( G/n ) =: E ( x ) . Thus, we get δe = ∂E∂x δx = − ∂E∂x (cid:20) ∂R∂x (cid:21) − ∂R∂L (cid:124) (cid:123)(cid:122) (cid:125) ∂r/∂L δL . (17)We observe that ∂E∂x = [0 0 ∂E∂x and that ∂E∂x < (asshown in Figure 3-left). Computing explicitly (17) we get ∂r∂L = µ µ µ + µ , (18)where µ = sω m ω u m ∂E∂x < µ = ω m ω g ω u − k u s ∂E∂x > µ = nω m ω g ω u ( ω m + m p ) > µ = L − npM > . (19) The latter inequality follows from npM = Lc × m M × M = m c L < L . Thus, from (18) and (19), we get ∂r∂L < for any L > .From the argument above we conclude that e = r ( L ) is strictly decreasing. This feature can be used to showcontraction of the reduced system. The reduced system andits linearization read ˙ L = φ ( r ( L )) − ω L Lδ ˙ L = ∂φ∂e (cid:124)(cid:123)(cid:122)(cid:125) > ∂r∂L (cid:124)(cid:123)(cid:122)(cid:125) < δL − ω L δL .δL .