Stress is an essential component of an organism s attempt

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1 Timing and bistability in the hypothalami-pituitary-adrenal axis: a model for onset and exposure therapy of stress disorders Lae Un Kim, Maria D Orsogna, Tom Chou Department of iomathematis, UCL, Los ngeles, C , Department of Mathematis, CalState-Northridge, Los ngeles, C 9, and Departments of iomathematis and Mathematis, UCL, Los ngeles, C Submitted to Proeedings of the National ademy of Sienes of the United States of meria The hypothalami-pituitary-adrenal (HP) axis is a neuroendorine system that regulates numerous physiologial proesses. This regulation is mediated by the inhibition of peptide hormones suh as ortiotropin-releasing hormone (CRH) and adrenoortiotropi hormone (CTH) by irulating gluoortioids suh as ortisol (CORT). Disruptions in the ativity of the HP axis are orrelated with many stress-related diseases suh as post-traumati stress disorder (PTSD) and major depressive disorder. In this paper, we hypothesize that normal and diseased states an be haraterized in terms of different basins of attration of a dynamial system. Our model distinguishes two omponents of negative feedbak by ortisol on irulating CRH levels: a slow diret suppression of CRH synthesis and a fast indiret effet on CRH release. Key physiologial features suh as ultradian osillations, CRH self-upregulation of CRH release, and external stressors are also inorporated. We show how these key ingredients in our model an yield two stable limit yles whih we assoiate with normal and diseased states. Transitions between the two an be indued by applying aute external stressors of appropriate intensity and duration. The ourrene of a transition is highly dependent upon the initiation time of external stress relative to the phase of the intrinsi ultradian osillation. Our results suggest that the timing of traumati events may be an important fator in determining if and how patients will exhibit hallmarks of PTSD. HP-axis PTSD Stress Disorders Dynamial system Stress is an essential omponent of an organism s attempt to adjust its internal state in response to environmental hange. The experiene, or even the pereption of physial and/or environmental hange, indues stress responses suh as the seretion of gluoortioids hormones (CORT) ortisol in humans and ortiosterone in rodents by the adrenal gland. The adrenal gland is one omponent of the hypothalamipituitary-adrenal (HP) axis, a group of interating endorine glands that play a entral role in stress response. The basi interations involving the HP axis are shown in Fig.. The paraventriular nuleus (PVN) of the hypothalamus reeives synapti inputs from various neural pathways via the entral nervous system that are ativated by both ognitive and physial stressors. One stimulated, CRH neurons in the PVN serete ortiotropin-releasing hormone (CRH), whih then stimulates the anterior pituitary gland to release adrenoortiotropin hormone (CTH) into the bloodstream. CTH then ativates a omplex signaling asade in the adrenal ortex, whih ultimately releases gluoortioids (Fig. ). In return, gluoortioids exert a negative feedbak on the hypothalamus and pituitary, suppressing CRH and CTH release and synthesis in an effort to return them to baseline levels. Classi stress responses inlude transient inreases in levels of CRH, CTH, and ortisol. The basi omponents and organization of the vertebrate neuroendorine stress axis arose early in evolution and the HP axis, in partiular, has been onserved aross mammals []. Fig.. Shemati of HP axis. () Stress is proessed in the entral nervous system (CNS) and a signal is relayed to the PVN in the hypothalamus to ativate CRH seretion into the hypophyseal portal system. () CRH diffuses to the pituitary gland and ativate CTH seretion. CTH travels down to the adrenal ortex to ativate ortisol (CORT) release. Cortisol inhibits both CRH and CTH seretion to down-regulate its own prodution, forming a losed loop. In the pituitary gland, ortisol binds to gluoortioid reeptors (GR) (yellow box) to inhibit CTH and self-upregulate GR prodution. This part of the axis omprises the P subsystem. (C) Negative feedbak of ortisol affets the synthesis proess in the hypothalamus, whih indiretly suppresses the release of CRH. External inputs suh as stressors and iradian inputs diretly affet the release rate of the CRH. Signifiane While orrelations between HP-axis dysregulation and psyhologial stress disorders suh as PTSD are known, to date, there is little mehanisti understanding of how stress response an ause altered ativity of the HP axis. y separating the synthesis and release proesses in CRH neurons in the hypothalamus and by inorporating a delayed response in the adrenal ortex, we model the HP axis as an exitable system with alternate stable osillating normal and diseased states. Transitions between the two osillating states an be indued by external stress of suffiiently long duration. Our mehanisti hypotheses provide insight into how stress response an alter HP axis dynamis and how re-exposure to stress an restore normal HP ativity. Reserved for Publiation Footnotes C PNS Issue Date Volume Issue Number

2 Mathematial models of the HP axis have been previously formulated in terms of dynamial systems of ordinary differential euations (ODEs) [,, 4, 5] or delay differential euations (DDEs) [6, 7, 8] that desribe the time-evolution of the key regulating hormones of the HP axis: CRH, CTH, and ortisol. These mathematial models [6, 7, 9] inlude the desription of positive self-regulation of gluoortioid reeptor expression in the pituitary, whih may generate bistability in the dynamial struture of the model []. Of the two stable euilibrium states, one is haraterized by higher levels of ortisol and is identified as the normal state. The other is haraterized by lower levels of ortisol and an be interpreted as one of the diseased states assoiated with hypoortisolism, as mentioned above. Stresses that affet the ativity of neurons in the PVN are desribed as perturbations to endogenous CRH seretion ativity. Depending on the length and magnitude of the stress input, the system may or may not shift from the basin of attration of the normal steady state towards that of the diseased one. If suh a transition does our, it may be interpreted as the onset of disease. later model [9] desribes the effet of stress on the HP axis as a gradual hange in the parameter values representing the maximum rate of CRH prodution and the strength of the negative feedbak ativity of ortisol. Here, hanges in ortisol levels are assumed to arise from anatomial hanges in the HP axis. oth lasses of models imply ualitatively different time ourses of disease progression [, 9]. The former suggests that the abnormal state is a pre-existing basin of attration of a dynamial model that stays dormant until a sudden transition is triggered by exposure to trauma []. In ontrast, the latter assumes that the abnormal state is reahed by the slow development of strutural hanges in physiology due to the traumati experiene [9]. lthough both models [, 9] desribe hanges in hormonal levels experiened by PTSD patients, they both fail to exhibit stable ultradian osillations in ortisol, whih, as we shall see, play an essential role in oupling HP axis dynamis to external stress inputs. In this study, we will demonstrate the funtional importane of a number of distintive physiologial features of the HP axis that have not been previously onsidered in mathematial models. These inlude, for example, the effets of intrinsi ultradian osillations on HP dysregulation, distint fast and slow feedbak mehanisms, and the orrelation between HP imbalane and disorders indued by external stress. First, as with the majority of hormones released by the body, ortisol levels undergo a iradian rhythm, starting low during night sleep, rapidly rising in the early morning, then gradually falling before rising again in the late afternoon. Superposed on this slow diurnal yle is an ultradian rhythm onsisting of approximately hourly pulses. CRH, CTH, and ortisol are all sereted episodially, with the pulses of CTH slightly preeding those of ortisol []. s for many other hormones suh as gonadotropin-releasing hormone (GnRH), insulin, and growth hormone (GH), the ultradian release pattern of gluoortioids is important in sustaining normal physiologial funtions, suh as regulating gene expression in the hippoampus []. It is unlear what role osillations play in homeostasis, but the time of onset of a stressor in relation to the phase of the ultradian osillation has been shown to influene the physiologial response eliited by the stressor []. seond property is the separation between the synthesis and release proess of CRH. CRH must first be synthesized before released by CRH neurons in the PVN (Fig. C). CRH release is mediated by synapti signals while synthesis reuires the up-regulation of transription and translation. Therefore, the two fundamentally different proesses of release and synthesis operate over very different timesales [4]. Finally, dysregulation in the HP axis is known to orrelate with a number of stress-related disorders. Inreased ortisol (hyperortisolism) is assoiated with major depressive disorder (MDD) [5, 6], while dereased ortisol (hypoortisolism) is a feature of posttraumati stress disorder (PTSD), post infetious fatigue, and hroni fatigue syndrome (CFS) [7, 8, 9, ]. Sine PTSD develops in the aftermath of extreme levels of stress experiened during traumati inidents like ombat, sexual abuse, or life-threatening aidents, its progression may be strongly orrelated with disruption of the HP axis aused by stress response. For example, lower peak and nadir ortisol levels were found in patients with ombat-related PTSD []. The mathematial model we derive inorporates the above physiologial features and reflets the basi physiology of the HP axis assoiated with delays in signaling, fast and slow negative feedbak mehanisms, and CRH self-upregulation. Depending on the parameters, our model may exhibit two distint stable osillating states. When two osillating states arise, one will have a larger osillation amplitude and a higher base ortisol level than the other. These two states will be interpreted as normal and diseased states. similar two-state dynamial struture arises in the lassi Fitzhugh-Nagumo model of a single neuron, in whih the resting and spiking states an be represented as bistable modes of the model [], and in models of neuronal networks where an epilepti brain is desribed in terms of the distane between a normal attrator and a seizure attrator in phase-spae []. ll of these systems, inluding the HP axis, an be driven to swith from one mode of ativity to another by hanges in system parameters or external stress inputs (Fig. ). Dynamial Model Previous models of HP dynamis [, 6, 9, 7, 4] are typially expressed in terms of ordinary differential euations (ODEs): dc =pci(t)fc(o) dc(c), [] d =pcf(or,o) d(), [] do =po(t) do(o), [] dr =prgr(or) dr(r), [4] where C(T), (T), and O(T) denote the plasma onentrations of CRH, CTH, and ortisol, respetively. R(T) represents the availability of gluoortioid reeptor (GR) in the anterior pituitary. Cortisol and ortisol-gr omplex are typially near euilibrium so its onentration is approximately proportional to the produt O(T)R(T) []. The parameters p α (α {C,,O,R}) relate the prodution rate of eah speies to speifi fators that regulate the rate of release/synthesis of the orresponding speies α. External stresses that drive CRH release by the PVN in the hypothalamus are represented by the input signal I(T). The funtion f C(O) desribes the negative feedbak of ortisol on CRH levels in the PVN while f (OR,O) desribes the negative feedbak of ortisol or ortisol-gr omplex (at onentration O(T)R(T)) in the pituitary. They are mathematially haraterized as being positive, dereasing funtions so that f,c( ) and f,c( ) <. On the other hand, the funtion g R(OR) desribes the self-upregulation effet of the ortisol- GR omplex on GR prodution in the anterior pituitary [5]. In ontrast to f,c( ), g R( ) is a positive but inreasing funtion of OR so that g R( ) and g R( ) >. Finally, the Footline uthor

3 degradation funtions d α( ) desribe how eah hormone and reeptor is leared and may be linear or nonlinear. Without inluding the effets of the gluoortioid reeptor (negleting E. 4 and assuming f (OR,O) = f (O) in E. ), Es. - form a rudimentary minimal model of the HP axis [, 6]. If f,c( ) are Hill-type feedbak funtions dependent only on O(T) and d α( ) are linear, a uniue global stable point exists. This euilibrium point transitions to a limit yle through a Hopf bifuration but only within nonphysiologial parameter regimes []. The inlusion of GR and its self-upregulation in the anterior pituitary [] reates two stable euilibrium states of the system, but still does not generate osillatory behavior. More reent studies extend the model to inlude nonlinear degradation [9] or onstant delay to aount for delivery of CTH and synthesis of gluoortioid in the adrenal gland [6]. These two extended models exhibit intrinsi iradian [9] or ultradian [6] osillations. However, both models permit only one osillating yle for any given set of parameter values [9, 6] preluding the mathematial distintion between normal and diseased states. Here, we develop a new model of the HP axis by first introduing a physiologially-motivated delay diretly into E.. This delay ultimately gives rise to the observed ultradian osillations [6]. We then distinguish the roles of slow diret CRH synthesis and fast CRH release in the negative feedbak of ortisol on CRH neurons. This allows us to separate the model into slow and fast omponents. Finally, self-upregulation of ortisol release is introdued whih allows for bistability in our model. These ingredients an be realistially ombined in a way that leads to novel, linially identifiable features and are systematially developed below. Ultradian rhythm and time delay -In order to apture the observed ultradian osillations in CTH and ortisol levels, we adapt a previous DDE model that inorporates a delayed feedforward and negative feedbak interation between the adrenal and pituitary glands, downstream of the hypothalamus. Indeed, experiments on rats show a -6 minute inherent delay in the response of the adrenal gland to CTH [7]. Moreover, in experiments performed on sheep [8], persistent ultradian osillations were observed even after surgially removing the hypothalamus, implying that osillations are inherent to the P subsystem. Sine osillations an be indued by delays, we assume, as in Walker et al. [6], a time delay T d in the CTHmediated ativation of ortisol prodution downstream of the hypothalamus. E. is thus modified to do = po(t T d) d OO. [5] Walker et al. [6] show that for fixed physiologial levels of CRH, the solution to Es., 4 and 5 leads to osillatory (T),O(T), and R(T). Synthesis vs. Release of CRH - CRH synthesis involves various pathways, inluding CRH gene transription and transport of pakaged CRH from the ell body (soma) to their axonal terminals where they are stored prior to release. Changes in these pathways and their effets on the overall rate of synthesis typially our on a timesale of minutes to hours. On the other hand, the seretory release proess depends on hanges in membrane potential at the axonal terminal of CRH neurons, whih our over milliseond to seond timesales. To model the synthesis and release proess separately, we distinguish two ompartments of CRH: the onentration of stored CRH within CRH neurons will be denoted C s(t), while levels of released CRH in the portal vein outside the neurons will be labeled C(T) (Fig. C). Newly synthesized CRH will first be stored, thus ontributing to C s. We assume that the stored CRH level C s relaxes toward a target value set by the funtion C (O): dc s = C (O) Cs T C. [6] Here, T C is a harateristi time onstant and C (O) is the ortisol-dependent target level of stored CRH (C (O)/T C an also be interpreted as the ortisol-dependent prodution rate). E. 6 also assumes that the relatively small amounts of CRH released into the bloodstream do not signifiantly deplete the C s pool. Previous studies have shown that variations in CRH gene expression due to hanges in ortisol levels take at least twelve hours to detet [9, ]. Therefore, we estimate T C hrs = 7min. The negative feedbak of ortisol on CRH levels thus ats through the prodution funtion C (O) on the relatively slow timesale T C. To motivate the funtional form of C (O), we rely experiments on rats with adrenal glands surgially removed [, 4]. Gluoortioid levels in these rats were then kept fixed (by injeting exogenous gluoortioid) for 5-7 days before intraellular CRH mrn levels were measured. The measured CRH mrn levels in the PVN were found to derease exponentially with the level of administered gluoortioid [, 4]. Thus, assuming the amount of releasable CRH is proportional to the amount of measured intraellular CRH mrn, we will approximate C (O) as a dereasing exponential funtion of ortisol level O. In ontrast to the inhibitory effet of gluoortioids on the synthesis proess, the seretory response of CRH neurons has been shown not to be diretly sensitive to gluoortioids under many types of stressors, inluding restraint and irulatory shok (hypovolemia) []. We limit our attention to these types of stressors and assume that the release of CRHfrom the intraellular pool is not diretly dependent on ortisol levels. Self-upregulation of CRH release -Finally, it has been hypothesized that CRH enhanes its own release [], espeially when external stressors are present. The enhanement of CRH release by CRH is mediated by ativation of the membrane-bound G-protein-oupled reeptor CRHR- whose downstream signaling pathways operate on timesales from milliseonds to seonds [4, 5]. Thus, self-upregulation of CRH release will be modeled by inluding a positive and inreasing funtion g C(C) in the soure term in E.. The seretion of CRH through synapti transmission-like mehanisms will also depend upon the amount of stored releasable CRH, C s(t), within the neuron and inside the synapti vesiles. Therefore, C s will also be fatored into E. through a soure term proportional to h(c s) whih is a funtion desribing the amount of CRH released per unit of ation potential ativity of CRH neurons (for example, I(t) desribes the overall firing rate of CRH neurons). Combining the two proposed self-upregulation proesses, we an rewrite E. by replaing f C(O) with h(c s)g C(C): dc = pci(t)h(cs)gc(c) dcc. [7] In this model, ortisol no longer diretly suppresses CRH levels, rather, it dereases CRH synthesis through E. 6, in turn suppressing C s. The ombination h(c s)g C(C) in E. 7 indiates the release rate of stored CRH dereases when either C s or C dereases. We assume that inputs into the CRH neurons modulate the overall release proess with weight p C. Footline uthor PNS Issue Date Volume Issue Number

4 Complete delay-differential euation model -We are now ready to inorporate the mehanisms desribed above into a new, more omprehensive mathematial model of the HP axis. In summary, our model inludes (i) delayed response of the adrenal ortex to ortisol (E. 5). (ii) slow time-sale negative feedbak by ortisol on CRH synthesis (through the C (O) prodution term in E. 6). (iii) fast-ating positive feedbak of stored and irulating CRH on CRH release (through the h(c s)g C(C) term in E. 7); The omplete mathematial model thus onsists of Es., 4, 5, 6, and 7. We heneforth assume f (OR,O) = f (OR) depends on only the ortisol-gr omplex and use speifi, previously used Hill-type funtions for f (OR) and g R(OR) [6, 7,, 9]. Our full theory is haraterized by the following system of delay differential euations: dc s =C (O) Cs, [8] T C dc =pci(t)h(cs)gc(c) dcc, [9] ( ) d =pc K d, [] K +OR do =po(t T d) d OO, [] ( ) dr µ RK =pr R d RR. [] KR +(OR) The parameters K,R represent the level of and R at whih the negative or positive effet are at their half maximum and µ R represents the basal prodution rate for GR when OR =. For a more detailed derivation and justifiation of Es. - and the hoie of the funtional forms, we refer the reader to [, 6]. Nondimensionalization - To simplify the further development and analysis of our model, we nondimensionalize our euations by resaling all variables and parameters in a manner similar to that of Walker et al. [6], as expliitly shown in the Supporting Information (SI). The dimensionless forms of Es. 8- beome d s dt = (o) s, t [] d = I(t)h(s)g(), dt [4] da dt = pa, +p (or) [5] do dt = a(t t d) o, [6] dr dt = (or) +p5 p6a, p 4 +(or) [7] where s,,a,r,o are the dimensionless versions of the original onentrations C s,c,,r,o, respetively. The dimensionless delay in ativation of ortisol prodution by CTH is now denoted t d. ll dimensionless parameters i,p i,t d, and t are ombinations of the physial parameters and are expliitly given in the SI. The funtions (o), h( s), and g () are dimensionless versions of C (O), h(c s), and g C(C), respetively, and will be hosen phenomenologially to be (o) = +e bo, h( s) = e ks, [8] µ g () = +( ) n. The form of (o) is based on the above-mentioned exponential relation observed in the experiments on adrenaletomized rats [, 4]. The parameters and b represent theminimum dimensionless level of stored CRH and the deay rate of the funtion, respetively. How the rate of CRH release inreases with s is given by the funtion h( s). Sine the amount of CRH pakaged in release vesiles is likely regulated, we assume h( s) saturates at high s. The hoie of a dereasing form for (o) implies that inreasing ortisol levels will derease the target level (or prodution rate) of s in E.. The redued prodution of s will then lead to a smaller h( s) and ultimately a redued release soure for (E. 4). s expeted, the overall effet of inreasing ortisol is a derease in the release rate of CRH. Finally, sine the upregulation of CRH release by irulating CRH is mediated by binding to CRH reeptor, g () will be hosen to be a Hill-type funtion, with Hill-exponent n, similar in form to the funtion g R(OR) used in Es. and 7. The parameter µ represents the basal release rate of CRH relative to the maximum release rate and represents the normalized CRH level at whih the positive effet is at half-maximum. Fast-slow variable separation and bistability - Sine we assume the negative feedbak effet of ortisol on synthesis of CRH operates over the longest harateristi timesale t in the problem, the full model must be studied aross two separate timesales, a fast timesale t, and a slow timesale τ = t/t εt. The full model (Es. -7) an be suintly written in the form d s = ε( (o) s), dt [9] dx = F(s,x), dt [] where x = (,a,o,r) is the vetor of fast dynamial variables, and F( s,x) denotes the right-hand-sides of Es We refer to the fast dynamis desribed by dx/dt = F( s,x) as a fast flow. In the ε limit, it is also easy to see that to lowest order s is a onstant aross the fast timesale and is a funtion of only the slow variable τ. Under this timesale separation, the first omponent of E. (E. 4) an be written as d = (s(τ),i)g(), [] dt where ( s(τ),i) Ih( s(τ)) = I( e ks(τ) ) is a funtion of s(τ) and I. Sine s is a funtion only of the slow timesale τ, an be viewed as a bifuration parameter ontrolling, over short timesales, the fast flow desribed by E.. One (t) uikly reahes its non-osillating uasieuilibrium value defined by d/dt = g () =, it an be viewed as a parametri term in E. 5 of the pituitaryadrenal (P) subsystem. Due to the nonlinearity of g (), the euilibrium value () satisfying g () = may be multi-valued depending on, as shown in Fig.. For ertain values of the free parameters, suh as n, µ, and, bistability an emerge through 4 Footline uthor

5 a saddle-node bifuration with respet to the bifuration parameter. Fig. shows the bifuration diagram, i.e., the nullline of defined by g () =. The dynamis of the P-subsystem depited in Fig. indiate the range of values that admit limit yle behavior for (a, o, r), while the fast -nullline depited in Fig. restrits the range of bistable values. Thus, bistable states that also support osillating (a,o,r) are possible only for values of that satisfy both riteria. Sine in the ε limit, irulating CRH only feeds forward into a,o, and r, a omplete desription of all the fast variables an be onstruted from just (E. 4 or ). Therefore, to visualize and approximate the dynamis of the full fivedimensional model, we only need to onsider the D projetion onto the fast and slow s variable. summary of the time-separated dynamis of the variables in our model is given in Fig. 4. g () inreasing L ( L, L ) L U U ( R, R ) Fig.. Nonlinear g () and bistability of fast variables. () The stable states of the deoupled system in E. an be visualized as the intersetion of the two funtions g () (dashed urve) and (gray line). For a given Hill-type funtion g (), E. an admit one or two stable states (solid irles), depending on funtion parameters. The unstable steady state is indiated by the open irle. () ifuration diagram of the deoupled system (E. ) with as the bifuration parameter. Solid and dashed segments represent stable and unstable steady states of the fast variables, respetively. L and U label basins of attration assoiated with the lower and upper stable branhes of the -nullline. Left and right bifuration points ( L, L ) and ( R, R ) are indiated. Fixed points of appear and disappear through saddle node bifurations as is varied through L and R. While the irulating CRH level reahes an euilibrium value defined by the -nullline, the P-subsystem an exhibit a limit yle in (a,o,r) when lies within a ertain range [6]. Values of (a,o,r) on the limit yle an be expressed as (a (θ;),o (θ;),r (θ;)), where θ = πt/t p() is the phase along the limit yle. a * () a * () physiologial regime 5 o* () o *() physiologial regime 5 Fig.. Dynamis of the osillating P-subsystem as a funtion of fixed. () Maximum/minimum and period-averaged values of CTH, a(t), as a funtion of irulating CRH. () Maximum/minimum and period-averaged values of ortisol o(t). Within physiologial CRH levels, CTH, GR, and ortisol osillate. The minima, maxima, and period-averaged ortisol levels typially inrease with inreasing. The plot was generated using dimensionless variables, a, and o with parameter values speified in [6] and t d =.44, orresponding to a delay of T d = 5min. slow variable ( ) τ in (t) D system (non osillating) fast variables x(t) a(t) o(t) r(t) P subsystem (osillating) Fig. 4. Classifiation of variables. Variables of the full five-dimensional model are grouped aording to their dynamial behavior. s(τ) is a slow variable, while x(t) = (,a,o,r) are fast variables. Of these, (a,o,r) form the typially osillatory P-subsystem that is reapitulated by. In the ε = /t limit, the variable s(τ) slowly relaxes towards a period-averaged value (o()). Therefore, the full model an be aurately desribed by its projetion onto the D ( s,) phase spae. To analyze the evolution of the slow variable s(τ), we write our euations in terms of τ = εt: d s dτ ε dx dτ = ( (o) s), [] = F(s,x). [] In the ε limit, the outer solution F( s,x) simply onstrains the system to be on the fast -nullline. The slow evolution of s(τ) along the fast -nullline depends on the value of the fast variable o(t) through (o). To lose the slow flow subsystem for s(τ), we fix to its euilibrium value as defined by the fast subsystem and approximate (o()) in E. by its period-averaged value () π (o (θ;)) dθ π = + π e bo (θ;)dθ π. [4] Sine o inreases with, () is a dereasing funtion of under physiologial parameter regimes. This period-averaging approximation allows us to relate the evolution of s(τ) in the slow subsystem diretly to. The evolution of the slow subsystem an be estimated by the losed D ( s,) system d s = () s, dτ [5] = h( s)i(t)g (). [6] y self-onsistently solving Es. 5 and 6, we an estimate trajetories of the full model when they are near the -nullline in the D ( s,)-subsystem. We will verify this in the following setion. Nullline struture and projeted dynamis- The separation of timesales results in a natural desription of the fast -nullline in terms of the parameter (Fig. ) and the slow s-nullline in terms of. However, the -nullline is plotted in the (,)- plane while the s-nullline is defined in the (, s)-plane. In order to plot the nulllines together, we relate the euilibrium value of s, (), to the oordinate through the monotoni relationship ( s) = Ih( () ) = I( e k () ) and transform the s variable into the diretion. Footline uthor PNS Issue Date Volume Issue Number 5

6 inreasing k N D shown in Fig. 6. Trajetories that start within the basin of attration of the lower stable branh of the -nullline ( initial state in Fig. 6) will stay on this branh for a long time before eventually sliding off near the bifuration point and jumping to the upper stable branh. Thus, the long-term behavior of our full model an be desribed in terms of the loations of the intersetions of nulllines of the redued system Fig. 5. Slow and fast nulllines and overall flow field. () The nullline of s in the ε limit is defined by s = (). These an be transformed into nulllines of through the relation = h( () ). For eah fixed value of, o(t;) is omputed by employing a built-in DDE solver dde in MTL. The numerial solution is then used to approximate () in E. 4 by Euler s method. The -nullline shifts to the right and gets steeper as k inreases. () The fast -nullline defined by g () = (blak urve) is plotted together with the slow -nullline (blue urve). Here, two intersetions arise orresponding to a high-ortisol normal (N) stable state and a low-ortisol diseased (D) stable state. The flow vetor field is predominantly aligned with the fast diretions toward the -nullline. Upon assuming a fixed basal stress input I =, these transformed nulllines the -nulllines are plotted in Fig. 5 for inreasing values of k, the parameter governing the sensitivity of CRH release to stored CRH. From the form h( () ) = ( e k () ), both the position and the steepness of the -nullline in (, )-spae depend strongly on k. Fig. 5 shows the two nulllines interseting at both stable branhes of the -nullline. The projeted D flows fields indiate that the trajetory is governed by fast flow over most of the (, )-spae. Clearly, how the fast and slow nulllines ross ontrols the long-term behavior of our model in the small ε limit. In general, the number of allowable nullline intersetions will depend on input level I and our parameters (,...,p 6,b,k,n,µ,t d ). Note that k, whih relates the amount of CRH inside the PVN neurons to its release rate in E. 4, mostly ontrols the loation and steepness of the slow -nullline. Other parameters suh as,, and µ appear diretly in the fast euation for and thus most strongly ontrol the fast -nullline. Fig. 6 shows that for a basal stress input of I = and an intermediate value of k, the nulllines ross at both stable branhes of the fast subsystem. s expeted, numerial simulations of our full model show the fast variables (a, o, r) uikly reahing their osillating states defined by the -nullline while the slow variable = Ih( s) remains fairly onstant. Independent of initial onfigurations that are nogt near the -nullline in (, )-spae, trajetories uikly jump to one of the stable branhes of the -nullline with little motion towards the -nullline, as indiated by ξ f. One near the -nullline, say when F( s,x) ε, the trajetories vary slowly aording to Es.. Here, the slow variable s relaxes to its steady state value while satisfying the onstraint F( s,x). In (,) spae, the system slowly slides along the-nullline towards the-nullline(theξ s paths in Fig. 6). This latter phase of the evolution ontinues until the system reahes an intersetion of the two nulllines, at whih the redued subsystem in s and reah euilibrium. If the fast variable is bistable, the two nulllines may interset at eah of the two stable branhes of the -nullline and yield the two distint stable solutions shown in Fig. 6. Depending on the parameters, suh as large k, the two nulllines may only interset on one stable branh of the -nullline as 5 5 -nullline -nullline trajetory initial state initial state ξ f ξ s ξ s ξ f initial state 5 5 -nullline -nullline trajetory initial state ξ f ξ s ξ s ξ f initial state Fig. 6. Euilibria at the intersetions of nulllines. () For intermediate values of k, there are three intersetions, two of them representing stable euilibria. Solid red lines are projetions of two trajetories of the full model, with initial states indiated by red dots and final stable states shown by blak dots. The full trajetories approah the intersetions of the -nullline (blue) and -nullline (blak). () For large k there is only one intersetion at the upper branh of the -nullline. Two trajetories with initial states near different branhes of the -nullline both approah the uniue intersetion ( ) on the upper branh. The senario shown here orresponds to a Type I nullline struture as desribed in the SI. Results and Disussion Changes in parameters that aompany trauma an lead to shifts in the position of the nulllines. For example, if the stored CRH release proess is suffiiently ompromised by trauma, the -nullline moves to the left, driving a bistable or fully resistant organism into a permanent diseased state. Interventions that inrease k would need to overome hysteresis in order to restore normal HP funtion. More permanent physiologial hanges suh as those resulting from traumati brain injury(ti) may derease the sensitivity of the pituitary to ortisol-gr omplex. This hange would be desribed by dereasing p in our model, leading to a leftward shift of the -nullline and an inreased likelihood of hypoortisolism. In this study, we shall fous on how external stress inputs an by themselves indue permanent but reversible transitions in HP dynamis without hanges in physiologi parameters. Speifially, we onsider only temporary hanges in I(t) and onsider the time-autonomous problem. Sine the majority of neural iruits that projet to the PVN are exitatory [7], we assume external stress stimulates CRH neurons to release CRH above its unit basal rate and that I(t) = +I ext(t) with I ext. To be onrete, we will estimate or onstrut many of the parameters from values used in previous studies, as listed in TableSintheSI.Ofthefourremainingparameters, µ,,, and k, we will study how our model depends on k while fixing µ,, and. Three possible nullline onfigurations arise aording to the values of µ,, and and are delineated in the SI. We have also impliitly onsidered only parameter regimes that yield osillations in the P subsystem at the stable states defined by the nullline intersetions. Given these onsiderations, we hose µ =.6, =.4, and = 77.8 for the rest of our analysis. This hoie of parameters is motivated in the SI and orresponds to a so- 6 Footline uthor

7 alled Type I nullline struture. In this ase, three possibilities arise: one intersetion on the lower stable branh of the -nullline if k < k L, two intersetions if k L < k < k R (Fig. 6), and one intersetion on the upper stable branh of the -nullline if k > k R (Fig. 6). For our hosen set of parameters and a basal stress input I =, the ritial values k L =.5 < k R =.54 are given by E. S in the SI. Normal stress response -tivation of the HP axis by aute stress ulminates in an inreased seretion of all three main hormones of the HP axis. Persistent hyperseretion may lead to numerous metaboli, affetive, and psyhoti dysfuntions [8, 9]. Therefore, reovery after stress-indued perturbation is essential to normal HP funtion. We explore the stability ofthehpaxisbyinitiatingthesystemintheupperofthetwo stable points shown in Fig. 7 and then imposing a min external stress input I ext =.. The HP axis responds with an inrease in the peak level of ortisol before relaxing bak to its original state after the stress is terminated (Fig. 7). This transient proess is depited in the projeted (, )-spae in Fig. 7. Upon turning on the stress, the lumped parameter and its nullline shift to the right by % sine = ( + I ext)h( () ). Thetrajetory will thenmoverapidlyupward towards the new value of on the -nullline; afterwards, it moves very slowly along the -nullline towards the shifted - nullline. fter min, the system arrives at the on the -nullline. When the stress is shut off, I =. I =, and the -nullline returns to its original position defined by I =. The trajetory also jumps bak to near (,) = (64.4,6) and subseuently uikly returns to the original upper-branh stable point stress, I ext = T (min) Fig. 7. Normal stress response. Numerial solution for the response to a min external stress I ext =.. () t the moment the external stress is turned on, the value of inreases from 64.4 to 7 and the irulating CRH level, uikly reahes its nullline before slowly evolving along it towards the slow -nullline. fter short durations of stress, the system returns to the normal state basin. () The peaks of the ortisol level are inreased during stress (red) but return to their original osillating values after the stress is turned off. External stress indues transition from normal to diseased state -Next, we disuss how transitions from a normal state to a diseased state an be indued by stress of suffiient duration. Suh transitions an be indued by a positive (exitatory) external stressor I ext of suffiient duration. Upon stimulation of the CRH neurons, both CRH and average gluoortioid levels are inreased while the average value of (o(t)) is dereased sine (o) is a dereasing funtion of o. s s(τ) slowly deaystowards thedereased target valueof (o()), h( s(τ)), and hene ( s), also derease. Muh of this derease ours along the stable branh of the -nullline. fter the external stress is swithed off, will jump bak down by a fator of /( + I ext). If the net derease in is suffiient to bring it below the bifuration value L, the system rosses the separatrix and approahes the alternate, diseased state. Thus, the normal-to-diseased transition is more likely to our if the external stress is maintained long enough to ause a large net derease in. The derease in inurred during the slow relaxation phase, plus the drop in assoiated with the essation of stress, represents the maximum possible redution in and most likely rossing of the upper separatrix. numerial solution of our model with a hr I ext =. was performed, and the trajetory in (, )-spae is shown in Fig. 8. The orresponding ortisol level along this trajetory is plotted in Fig. 8, showing that indeed a permanent transition to the lower ortisol state ourred shortly after the essation of stress. C stress stress I ext =. I ext =. D T (min) Fig. 8. Stress-indued transitions into an osillating lowortisol diseased state. n exitatory external stress I ext =. is applied for hrs. Here, the system reahes the new stable point set by I =. before stress is terminated and the -nullline reverts to its original position set by I =. () t intermediate values of.5 < k <.54, when two stable state arise, a transition from the normal high-ortisol state into the diseased low-ortisol state an be indued by hroni external stress. () Numerial solutions of ortisol level plotted against the original time variable T shows the transition to the low-ortisol diseased state shortly after essation of stress. (C) and (D) If k > k R =.54, only the normal stable state exists. The system will reover and return to its original healthy state after a transient period of low ortisol. In addition to a long-term external stress, the permanent transition to a diseased state reuires.5 < k <.54 and the existene of two stable points. On the other hand, when k > k R =.54, the enhaned CRH release stimulates enough ortisol prodution to drive the sole long term solution to the stable upper normal branh of the -nullline, rendering the HP system resistant to stress-indued transitions. The response to hroni stress initially follows the same pattern as desribed above for the two-stable-state ase, as shown in Fig. 8C. However, the system will ontinue to evolve along the lower branh towards the -nullline, eventually sliding off the lower branh near the right bifuration point ( R, R) and returning to the single normal euilibrium state. Thus, when k is suffiiently high, the system may experiene a transient period of lowered ortisol level after hroni stress but will Footline uthor PNS Issue Date Volume Issue Number 7

8 eventually reover and return to the normal ortisol state. The orresponding ortisol level shown in Fig. 8D shows this reovery at T min, whih ours approximately 5min after the essation of stress T (min) 5 5 C stress, I ext = T (min) D stress, I ext = Fig. 9. Stress timing and transition to low-ortisol osillating state. Cortisol levels in response to I ext =. applied over 5min. () If stress is initiated at T = 5min, a transition to the low-ortisol diseased state is triggered. () If stress is initiated at T = min, the system returns to its normal high-ortisol state. Note that the first peak (marked by ) during the stress in () is higher than the first peak in (). (C) If stress is initiated at T = 5min 5min, stress essation and the slow relaxation along the -nullline during stress are suffiient to bring just left of the separatrix, induing the transition. (D) For initiation time T = min, remains to the right of the separatrix, preluding the transition. Transition to diseased state depends on stress timing -We have shown how transitions between the osillating normal and diseased states depend on the duration of the external stress I ext. However, whether a transition ours also depends on the time, relative to the phase of the intrinsi ultradian osillations, at whih a fixed-duration external stress is initiated. Toillustrate thisdependeneonphase, weplotinfigs. 9and two solutions for obtained with a 5min I ext =. initiated at different phases of the underlying ortisol osillation. If stress is initiated during the rising phase of the osillations, a transition to the low-ortisol diseased state ours and is ompleted at approximately T = min (Fig. 9,C). If, however, stress is initiated during the falling phase, the transition does not our and the system returns to the normal stable state (Fig. 9,D). In this ase, a longer stress duration would be reuired to push the trajetory past the low- separatrix into the diseased state. s disussed earlier, an inrease in period-averaged ortisol level during stress drives a normal-to-diseased transition. We see that the period-averaged level of ortisol during stress is different for stress started at min from stress started at 5min. s detailed in the SI, the amplitude of the first ortisol peak after the start of stress is signifiantly lower when the stress starts during the falling phase of the intrinsi ortisol osillations. The differene between initial responses in o(t) affets the period-averaging in (o) during external stress, ultimately influening s and onseuently determining whether or not a transition ours. Note that this phase dependene is appreiable only when the stress duration is near the threshold value that brings the system lose to the separatrix between normal and diseased basins of attration. Trajetories near the separatrix are sensitive to small hanges in the negative feedbak of ortisol on CRH synthesis, suh as those hanges aused by varying the timing of the stress signal. Stress of intermediate duration an indue reverse transitions - Our theory also allows the possibility for positive stressors I ext to indue reverse transitions from the diseased to the normal state. Understanding these reverse transitions is useful in the ontext of exposure therapy (ET), where PTSD patients are subjeted to stressors in a ontrolled and safe manner, using for example, omputer-simulated virtual reality exposure. Within our model, upon applying stress and starting from the low- diseased euilibrium point, the horizontal shift in resultingfrom ET (I ext > ) auses thesystem to move rightward aross the separatrix and suggests a transition to the high- normal state an our. C stress, I ext =. stress I D ext = T (min) Fig.. Stress-indued transitions to high-ortisol osillating state. () Projeted D system dynamis when a stressor of amplitude I ext =. is applied for 9min starting at T = min. is inreased just above the unstable branh ( ) to allow the unstressed system to ross the separatrix and transition to the normal high- stable state. () The plot of shows the transition to the high-ortisol, high-osillation amplitude state shortly after the 9min stress. (C) stressor turned off after 78min (hrs) leaves the system in the basin of attration of the diseased state. (D) Cortisol levels are pushed up but after about min relax bak to levels of the original diseased state. s shown in Fig., if the stressor is applied long enough, the trajetory reahes a point above the unstable branh of the -nullline upon termination and the system will reah the normal, high-ortisol state (Fig. ). Sine the initial motion is governed by fast flow, the minimum stress duration needed to inite the diseased-to-normal transition is short (on the timesale of minutes). However, if the stressor is applied for too long, a large redution in is experiened along the upper stable branh. Cessation of stress might then lower bak into the basin of attration of the low-ortisol diseased 8 Footline uthor

9 state (Fig. C). Fig. D shows the ortisol level transiently inreasing to a normal level before reverting bak to low levels after approximately min. Within our dynamial model, stresses need to be of intermediate duration in order to indue a permanent transition from the diseased to the normal state. Its ourrene may also depend on the phase (relative to the intrinsi osillations of the fast P subsystem) over whih stress was applied, espeially when the stress duration is near its transition thresholds. For a reverse diseased-to-normal transition to our, the derease in s annot be so large that it brings the trajetory past the left separatrix, as shown in Fig. C. Therefore, near the maximum duration, stress initiated over the falling phase of ortisol osillation will be more effetive at triggering the transition to a normal high-ortisol state. Overall, these results imply that exposure therapy should be arefully tuned in order to optimally drive the dynamis of the HP axis to a normal state in patients with hypoortisolism-assoiated stress disorders. Summary and Conlusions We developed a theory of HP dynamis that inludes stored CRH, irulating CRH, CTH, ortisol and gluoortioid reeptor. Our model inorporates a fast self-upregulation of CRH release, a slow negative feedbak effet of ortisol on CRH synthesis, and a delay in CTH-ativated ortisol synthesis. These ingredients allow our model to be separated into slow and fast omponents and projeted on a D subspae for analysis. Depending on physiologial parameters, there may exist zero, one, or two stable simultaneous solutions of both fast and slow variables. We hose to fix the other parameters and analyze our model as a funtion of k, the parameter haraterizing the dependene of CRH seretion on stored CRH levels. For small k, CRH release is weakened and only the low-crh euilibrium point arises; suh an organism is trapped in the low-ortisol diseased state. For large k, only the high-crh normal state arises, rendering the organism resistant to auiring the long-term, low-ortisol side-effet of ertain stress disorders. When only one stable solution arises, HP dysregulation must depend on hanges in parameters resulting from permanent physiologial modifiations due to e.g., aging, physial trauma, or stress itself [9, ]. For example, it has been observed that older rats exhibit inreased CRH seretion while maintaining normal levels of CRH mrn in the PVN [4]. Suh a hange ould be interpreted as an age-dependent inrease in k, whih, in our model, implies that aging makes the organism more resistant to stress-indued hypoortisolism. Indeed, it has been suggested that prevalene of PTSD delines with age [4, 4]. Within ertain parameter regimes and for intermediate k, our theory an also exhibit bistability. When two stable solutions arise, we identify the states with low osillating levels of ortisol as the diseased state assoiated with hypoortisolism. Transitions between different stable states an be indued by temporary external stress inputs, implying that HP dysregulation may develop without permanent strutural or physiologial hanges. Stresses that affet seretion of CRH by the PVN are shown to be apable of induing transitions from normal to diseased states provided they are of suffiient duration (Fig. 8). Our model offers a mehanisti explanation to the seemingly ounter-intuitive phenomenon of lower ortisol levels after stress-indued ativation of ortisol prodution. Solutions to our model demonstrate that the negative-feedbak effet of a temporary inrease in ortisol on the synthesis proess of CRH an slowly aumulate during the stress response and eventually shift the system into a different basin of attration. Suh a mehanism provides an alternative to the hypothesis that hypoortisolism in PTSD patients results from permanent hanges in physiologial parameters assoiated with a negative-feedbak mehanism [44, 45]. We also find that external stress an indue the reverse transition from a diseased low-ortisol state to the normal high-ortisol state. Our results imply that re-exposure to stresses of intermediate duration an drive the system bak to normal HP funtion, possibly deoupling stress disorders from hypoortisolism. Interestingly, we show that the minimum durations reuired for either transition depends on the time at whih the stress is initiated relative to the phase of the intrinsi osillations in (a, o, r). Due to subtle differenes in ortisol levels immediately following stress initiation at different phases of the intrinsi ortisol osillation, the different umulative negativefeedbak effet on CRH an determine whether or not a trajetory rosses a separatrix (Fig. 9). When the duration of external stress is near its threshold, normal-to-diseased state transitions are easier to indue when stress is initiated during the rising phase of ortisol osillations. Reverse diseasedto-normal transitions are more easily indued when stress is initiated during the falling phase. In total, our theory provides a mehanisti piture that onnets ortisol dysregulation with stress disorders and a mathematial framework one an use to study the downstream effets of therapies suh as brief eleti psyhotherapy (EP) and exposure therapy (ET). oth therapies involve reexperiening stressful situations diretly or through imagination, and have been onsistently proven effetive as first-line treatments for PTSD symptoms [46, 47, 48]. HP ativity under more omplex therapy protools an be tested via simulation of our model by using the appropriate external input funtion I ext(t). It is important to emphasize that we modeled neuroendorine dynamis downstream of the stress input I ext. How the form of the stress funtion I ext depends on the type of stress experiened reuires a more detailed study of more upstream proesses, inluding how hormones might feedbak to these upstream proesses. Sine higher ortisol levels are found among female PTSD patients with a history of hildhood abuse [49] and among PTSD patients who have experiened a nulear aident [5], future studies of suh divergent, experiene-dependent dysregulation will rely on more omplex input funtions I ext(t), parameter hanges, and memory effets. Many other novel features of our model remain to be explored. For example, under periodi driving, omplex resonant behavior should arise depending on the amplitude and freueny of the external stress I ext(t) and the nullline struture of the speifi system. knowledgments This work was supported by the rmy Researh Offie via grant W9NF and the NSF through grant CS- 48. The authors also thank professors T. Minor and M. Wehselberger for insightful disussions. Footline uthor PNS Issue Date Volume Issue Number 9