Onset, timing, and exposure therapy of stress disorders: mechanistic insight from a mathematical model of oscillating neuroendocrine dynamics

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1 Kim et al. RESEARCH arxiv: v [q-bio.nc] 9 Mar 26 Onset, timing, and exposure therapy of stress disorders: mehanisti insight from a mathematial model of osillating neuroendorine dynamis Lae Kim, Maria D Orsogna and Tom Chou * * Correspondene: tomhou@ula.edu 3 Dept. of Biomathematis, Univ of California, Los Angeles, Los Angeles, USA Full list of author information is available at the end of the artile Equal ontributor Abstrat The hypothalami-pituitary-adrenal (HPA) axis is a neuroendorine system that regulates numerous physiologial proesses. Disruptions in the ativity of the HPA axis are orrelated with many stress-related diseases suh as post-traumati stress disorder (PTSD) and major depressive disorder. In this paper, we haraterize normal and diseased states of the HPA axis as basins of attration of a dynamial system desribing the inhibition of peptide hormones suh as ortiotropin-releasing hormone (CRH) and adrenoortiotropi hormone (ACTH) by irulating gluoortioids suh as ortisol (CORT). In addition to inluding key physiologial features suh as ultradian osillations in ortisol levels and self-upregulation of CRH neuron ativity, our model distinguishes the relatively slow proess of ortisol-mediated CRH biosynthesis from rapid trans-synapti effets that regulate the CRH seretion proess. Cruially, we find that the slow regulation mehanism mediates external stress-driven transitions between the stable states in novel, intensity, duration, and timing-dependent ways. These results indiate that the timing of traumati events may be an important fator in determining if and how patients will exhibit hallmarks of stress disorders. Our model also suggests a mehanism whereby exposure therapy of stress disorders suh as PTSD may at to normalize downstream dysregulation of the HPA axis. Keywords: HPA-axis; PTSD; Stress Disorders; Dynamial system Introdution 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 hypothalami-pituitary-adrenal (HPA) axis, a olletion of interating neuroendorine ells and endorine glands that play a entral role in stress response. The basi interations involving the HPA 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

2 Kim et al. Page 2 of 3 ortiotropin-releasing hormone (CRH), whih then stimulates the anterior pituitary gland to release adrenoortiotropin hormone (ACTH) into the bloodstream. ACTH then ativates a omplex signaling asade in the adrenal ortex, whih ultimately releases gluoortioids (Fig. B). In return, gluoortioids exert a negative feedbak on the hypothalamus and pituitary, suppressing CRH and ACTH release and synthesis in an effort to return them to baseline levels. Classi stress responses inlude transient inreases in levels of CRH, ACTH, and ortisol. The basi omponents and organization of the vertebrate neuroendorine stress axis arose early in evolution and the HPA axis, in partiular, has been onserved aross mammals []. Figure : Shemati of HPA axis. (A) 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. (B) CRH diffuses to the pituitary gland and ativate ACTH seretion. ACTH travels down to the adrenal ortex to ativate ortisol (CORT) release. Cortisol inhibits both CRH and ACTH seretion to downregulate its own prodution, forming a losed loop. In the pituitary gland, ortisol binds to gluoortioid reeptors (GR) (yellow box) to inhibit ACTH and self-upregulate GR prodution. This part of the axis omprises the PA 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. Dysregulation in the HPA axis is known to orrelate with a number of stressrelated disorders. Inreased ortisol (hyperortisolism) is assoiated with major depressive disorder (MDD) [2, 3], while dereased ortisol (hypoortisolism) is a feature of post-traumati stress disorder (PTSD), post infetious fatigue, and hroni fatigue syndrome (CFS) [4 7]. 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 HPA axis aused by stress response. For example, lower peak and nadir ortisol levels were found in patients with ombat-related PTSD [8].

3 Kim et al. Page 3 of 3 Mathematial models of the HPA axis have been previously formulated in terms of dynamial systems of ordinary differential equations (ODEs) [9 2] or delay differential equations (DDEs) [3 5] that desribe the time-evolution of the key regulating hormones of the HPA axis: CRH, ACTH, and ortisol. These models [3, 4, 6] inorporate positive self-regulation of gluoortioid reeptor expression in the pituitary, whih may generate bistability in the dynamial struture of the model [7]. Of the two stable equilibrium 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. 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. A later model [6] desribes the effet of stress on the HPA 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. Changes in ortisol seretion pattern are assumed to arise from anatomial hanges that are mathematially represented as hanges to the orresponding parameter values [6]. Both lasses of models imply qualitatively different time ourses of disease progression [6, 7]. 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 [7]. 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 [6]. Although both models [6, 7] desribe hanges in hormonal levels experiened by PTSD patients, they both fail to exhibit stable ultradian osillations in ortisol, whih is known to play a role in determining the responsiveness of the HPA axis to stressors [8]. In this study, we onsider a number of distintive physiologial features of the HPA axis that give a more omplete piture of the dynamis of stress disorders and that have not been onsidered in previous mathematial models. These inlude the effets of intrinsi ultradian osillations on HPA dysregulation, distint rapid and slow feedbak ations of ortisol, and the orrelation between HPA imbalane and disorders indued by external stress. As with the majority of hormones released by the body, ortisol levels undergo a iradian rhythm, starting low during night sleep, rapidly rising and reahing its peak in the early morning, then gradually falling throughout the day. Superposed on this slow diurnal yle is an ultradian rhythm onsisting of approximately hourly pulses. CRH, ACTH, and ortisol are all sereted episodially, with the pulses of ACTH slightly preeding those of ortisol [9]. As 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 [2]. 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 [2].

4 Kim et al. Page 4 of 3 To distinguish the rapid and slow ations of ortisol, we separate the dynamis of biosynthesis of CRH from its seretion proess, whih operate over very different timesales [22]. While the two proesses are mostly independent from eah other, the rate of CRH seretion should depend on the synthesis proess sine CRH peptides must be synthesized first before being released (Fig. C). On the other hand, the rate of CRH peptide synthesis is influened by ortisol levels, whih in turn, are regulated by released CRH levels. We will investigate how the separation and oupling of these two proesses an allow stress-indued dysregulations of the HPA axis. The mathematial model we derive inorporates the above physiologial features and reflets the basi physiology of the HPA axis assoiated with delays in signaling, fast and slow negative feedbak mehanisms, and CRH self-upregulation [23]. Within an appropriate parameter regime, our model exhibits two distint stable osillating states, of whih one is marked by a larger osillation amplitude and a higher base ortisol level than the other. These two states will be referred to as normal and diseased states. Our interpretation is reminisent of the two-state dynamial struture that arises in the lassi Fitzhugh-Nagumo model of a single neuron, in whih resting and spiking states emerge as bistable modes of the model [24], or in models of neuronal networks where an epilepti brain is desribed in terms of the distane between a normal and a seizure attrator in phase-spae [25]. Models Models of HPA dynamis [3, 4, 6, 7, 26] are typially expressed in terms of ordinary differential equations (ODEs): dc dt =p CI(T )f C (O) d C (C), () da dt =p ACf A (OR, O) d A (A), (2) do dt =p OA(T ) d O (O), (3) dr dt =p Rg R (OR) d R (R), (4) where C(T ), A(T ), and O(T ) denote the plasma onentrations of CRH, ACTH, and ortisol at time T, respetively. R(T ) represents the availability of gluoortioid reeptor (GR) in the anterior pituitary. The amount of ortisol bound GR is typially in quasi-equilibrium so onentration of the ligand-reeptor omplex is approximately proportional to the produt O(T )R(T ) [7]. The parameters p α (α {C, A, O, R}) relate the prodution rate of eah speies α to speifi fators that regulate the rate of release/synthesis. 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 A (OR, O) desribes the negative feedbak of ortisol or ortisol-gr omplex (at onentration O(T )R(T )) in the pituitary. Both are mathematially haraterized as being positive, dereasing funtions so that f A,C ( ) and f A,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 [27]. In ontrast

5 Kim et al. Page 5 of 3 to f A,C ( ), g R ( ) is a positive but inreasing funtion of OR so that g R ( ) and g R ( ) >. Finally, the 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 Eq. 4 and assuming f A (OR, O) = f A (O) in Eq. 2), Eqs. -3 form a rudimentary minimal model of the HPA axis [9, 28]. If f A,C ( ) are Hill-type feedbak funtions dependent only on O(T ) and d α ( ) are linear, a unique global stable point exists. This equilibrium point transitions to a limit yle through a Hopf bifuration but only within nonphysiologial parameter regimes [9]. The inlusion of GR and its selfupregulation in the anterior pituitary [7] reates two stable equilibrium states of the system, but still does not generate osillatory behavior. More reent studies extend the model (represented by Eq. -4) to inlude nonlinear degradation [6] or onstant delay to aount for delivery of ACTH and synthesis of gluoortioid in the adrenal gland [3]. These two extended models exhibit only one intrinsi iradian [6] or ultradian [3] osillating yle for any given set of parameter values, preluding the interpretation of normal and diseased states as bistable osillating modes of the model. Here, we develop a new model of the HPA axis by first adapting previous work [3] where a physiologially-motivated delay was introdued into Eq. 3, giving rise to the observed ultradian osillations [3]. We then improve the model by distinguishing the relatively slow mehanism underlying the ortisol-mediated CRH biosynthesis from the rapid trans-synapti effets that regulate the CRH seretion proess.this allows us to deompose the dynamis into slow and fast omponents. Finally, self-upregulation of CRH release is introdued whih allows for bistability. 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 Experiments on rats show a 3-6 minute inherent delay in the response of the adrenal gland to ACTH [29]. Moreover, in experiments performed on sheep [3], persistent ultradian osillations were observed even after surgially removing the hypothalamus, implying that osillations are inherent to the PA subsystem. Sine osillations an be indued by delays, we assume, as in Walker et al. [3], a time delay T d in the ACTH-mediated ativation of ortisol prodution downstream of the hypothalamus. Eq. 3 is thus modified to do dt = p OA(T T d ) d O O. (5) Walker et al. [3] show that for fixed physiologial levels of CRH, the solution to Eqs. 2, 4 and 5 leads to osillatory A(T ), O(T ), and R(T ). In order to desribe the observed periodi ortisol levels in normal and diseased states, the model requires two osillating stable states. We will see that dual osillating states arise within our model when the delay in ACTH-mediated ativation of ortisol prodution is oupled with other known physiologial proesses.

6 Kim et al. Page 6 of 3 Synthesis 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 the steady state of the synthesis proess 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 dt = C (O) C s T C. (6) Here, T C is a harateristi time onstant and C (O) is the ortisol-dependent target level of stored CRH. Eq. 6 also assumes that the relatively small amounts of CRH released into the bloodstream do not signifiantly deplete the C s pool. Note that the effets indued by hanging ortisol levels are immediate as the prodution term C (O)/T C is adjusted instantaneously to urrent ortisol levels. Our model thus does not exlude ortisol rapidly ating on the initial transription ativity, as suggested by CRH hnrna (preursor mrna) measurements [3]. On the other hand, the time required to reah the steady state for the ompletely synthesized CRH peptide will depend on the harateristi time sale onstant T C. Ideally, T C should be estimated from measurements of the pool size of releasable CRH at the axonal terminals. To best of our knowledge, there are urrently no suh measurements available, so we base our estimation on mrna level measurements. We believe this is a better representation of releasable CRH than hnrna levels sine mrna synthesis is a further downstream proess. Previous studies have shown that variations in CRH mrna due to hanges in ortisol levels take at least twelve hours to detet [32]. Therefore, we estimate T C 2hrs = 72min. 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 invoke experiments on rats whose adrenal glands had been surgially removed and in whih gluoortioid levels were subsequently kept fixed (by injeting exogenous gluoortioid) for 5-7 days [22, 33]. The measured CRH mrna levels in the PVN were found to derease exponentially with the level of administered gluoortioid [22, 33]. Assuming the amount of releasable CRH is proportional to the amount of measured intraellular CRH mrna, we an approximate C (O) as a dereasing exponential funtion of ortisol level O. Seretion of CRH To desribe the CRH seretion, we onsider the following three fators: synapti inputs to CRH ells in the PVN, availability of releasable CRH peptide, and selfupregulation of CRH release.

7 Kim et al. Page 7 of 3 CRH seretion ativity is regulated by synapti inputs reeived by the PVN from multiple brain regions inluding limbi strutures like the hippoampus and the amygdala, that are ativated during stress. It has been reported that for ertain types of stressors, these synapti inputs are modulated by ortisol independent of, or parallel to, its regulatory funtion on CRH synthesis ativity [34]. On the other hand, a series of studies [35 37] showed that ortisol did not affet the basal spiking ativity of the PVN. We model the overall synapti input, denoted by I(T ) in Eq., as follows I(T ) = I base + I ext (T ), (7) where I base and I ext (T ) represent the basal firing rate and stress-dependent synapti input of the PVN, respetively. As the effet of ortisol on the synapti input during stress is speifi to type of stressor [38 4], we assume I ext (T ) to be independent of O for simpliity and generality. Possible impliations of ortisol dependent input funtion I ext (T, O) on model behavior will be disussed in the Additional File. The seretion of CRH will also depend upon the amount of stored releasable CRH, C s (T ), within the neuron and inside the synapti vesiles. Therefore, C s an also be fatored into Eq. through a soure term h(c s ) whih desribes the amount of CRH released per unit of ation potential ativity of CRH neurons. Finally, it has been hypothesized that CRH enhanes its own release [23], 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, 42]. Thus, self-upregulation of CRH release an be modeled by inluding a positive and inreasing funtion g C (C) in the soure term in Eq.. Combining all these fators involved in regulating the seretion proess, we an rewrite Eq. by replaing f C (O) with h(c s )g C (C) as follows dc dt = p CI(T )h(c s )g C (C) d C C. (8) In this model (represented by Eqs. 6,8,2,5, and 4), ortisol no longer diretly suppresses CRH levels, rather, it dereases CRH synthesis through Eq. 6, in turn suppressing C s. The ombination h(c s )g C (C) in Eq. 8 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. Complete delay-differential equation model We are now ready to inorporate the mehanisms desribed above into a new, more omprehensive mathematial model of the HPA axis, whih, in summary, inludes (i) A delayed response of the adrenal ortex to ortisol (Eq. 5). (ii) A slow time-sale negative feedbak by ortisol on CRH synthesis (through the C (O) prodution term in Eq. 6). (iii) A fast-ating positive feedbak of stored and irulating CRH on CRH release (through the h(c s )g C (C) term in Eq. 8);

8 Kim et al. Page 8 of 3 Our omplete mathematial model thus onsists of Eqs. 2, 4, 5, 6, and 8. We heneforth assume f A (OR, O) = f A (OR) depends on only the ortisol-gr omplex and use Hill-type funtions for f A (OR) and g R (OR) [3, 4, 6, 7]. Our full theory is haraterized by the following system of delay differential equations: dc s dt =C (O) C s, (9) T C dc dt =p CI(T )h(c s )g C (C) d C C, () ( ) da dt =p K A AC d A A, () K A + OR do dt =p OA(T T d ) d O O, (2) ( dr dt =p µ R K 2 ) R R KR 2 + d R R. (3) (OR)2 The parameters K A,R represent the level of A 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 =. Of all the proesses modeled, we will see that the slow negative feedbak will be ruial in mediating transitions between stable states of the system. The slow dynamis will allow state variables to ross basins of attration assoiated with eah of the stable states. Nondimensionalization To simplify the further development and analysis of our model, we nondimensionalize Eqs. 9-3 by resaling all variables and parameters in a manner similar to that of Walker et al. [3], as expliitly shown in the Additional File. We find d s dt = (o) s, t (4) d dt = q I(t)h( s )g () q 2, (5) da dt = + p 2 (or) p 3a, (6) do dt = a(t t d) o, (7) dr dt = (or)2 p 4 + (or) 2 + p 5 p 6 a, (8) where s,, a, r, o are the dimensionless versions of the original onentrations C s, C, A, R, O, respetively. The dimensionless delay in ativation of ortisol prodution by ACTH is now denoted t d. All dimensionless parameters q i, p i, t d, and t are ombinations of the physial parameters and are expliitly given in the Additional File. 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

9 Kim et al. Page 9 of 3 (o) = + e bo, h( s ) = e ks, (9) µ g () = + (q ) n. The form of (o) is based on the above-mentioned exponential relation observed in adrenaletomized rats [22, 33]. The parameters and b represent the minimum 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 Eq. 4. The redued prodution of s will then lead to a smaller h( s ) and ultimately a redued release soure for (Eq. 5). As 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 Eqs. 3 and 8. The parameter µ represents the basal release rate of CRH relative to the maximum release rate and q 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 (Eqs. 4-8) an be suintly written in the form d s dt = ε( (o) s ), (2) dx dt = F( s, x), (2) where x = (, a, o, r) is the vetor of fast dynamial variables, and F( s, x) denotes the right-hand-sides of Eqs 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 Eq. 2 (Eq. 5) an be written as d dt = q( s(τ), I)g () q 2, (22) where q( s (τ), I) q Ih( s (τ)) = q I( e ks(τ) ) is a funtion of s (τ) and I. Sine s is a funtion only of the slow timesale τ, q an be viewed as a bifuration

10 Kim et al. Page of 3 parameter ontrolling, over short timesales, the fast flow desribed by Eq. 22. One (t) quikly reahes its non-osillating quasi-equilibrium value defined by d/dt = qg () q 2 =, it an be viewed as a parametri term in Eq. 6 of the pituitaryadrenal (PA) subsystem. Due to the nonlinearity of g (), the equilibrium value (q) satisfying qg () = q 2 may be multi-valued depending on q, as shown in Figs. 2A and 2B. For ertain values of the free parameters, suh as n, µ, and q, bistability an emerge through a saddle-node bifuration with respet to the bifuration parameter q. Fig. 2B shows the bifuration diagram, i.e., the nullline of defined by qg () = q 2. q 2 qg () L U (q L, L ) U inreasing q (q R, R ) A A L q B Figure 2: Nonlinear g () and bistability of fast variables. (A) The stable states of the deoupled system in Eq. 22 an be visualized as the intersetion of the two funtions qg () (dashed urve) and q 2 (gray line). For a given Hill-type funtion g (), Eq. 22 an admit one or two stable states (solid irles), depending on funtion parameters. The unstable steady state is indiated by the open irle. (B) Bifuration diagram of the deoupled system (Eq. 22) with q 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 (q L, L) and (q R, R) are indiated. Fixed points of appear and disappear through saddle node bifurations as q is varied through q L and q R. For equilibrium values of lying within a ertain range, the PA-subsystem an exhibit a limit yle in (a, o, r) [3] that we express as (a (θ; ), o (θ; ), r (θ; )), where θ = 2πt/t p () is the phase along the limit yle. The dynamis of the PAsubsystem depited in Fig. 3 indiate the range of values that admit limit yle behavior for (a, o, r), while the fast -nullline depited in Fig. 2B 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 whih obeys Eq. 22. Therefore, to visualize and approximate the dynamis of the full fivedimensional model, we only need to onsider the 2D projetion onto the fast and slow s variable. A summary of the time-separated dynamis of the variables in our model is given in Fig. 4.

11 Kim et al. Page of 3 3 a * () a * () A 3 o* () o *() B 2 2 physiologial regime physiologial regime Figure 3: Dynamis of the osillating PA-subsystem as a funtion of fixed. (A) Maximum/minimum and period-averaged values of ACTH, a(t), as a funtion of irulating CRH. (B) Maximum/minimum and period-averaged values of ortisol o(t). Within physiologial CRH levels, ACTH, GR (not shown), 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 [43] and t d =.44, orresponding to a delay of T d = 5min. slow variable fast variables x(t) ( ) τ s (t) a(t) o(t) r(t) 2D system (non osillating) PA subsystem (osillating) Figure 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 PA-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 2D ( s, ) phase spae. To analyze the evolution of the slow variable s (τ), we write our equations in terms of τ = εt: d s dτ = ( (o) s ), (23) ε dx dτ = F( s, x). (24) In the ε limit, the outer solution F( s, x) simply onstrains the system to be on the fast -nullline defined by qg () = q 2. The slow evolution of s (τ) along

12 Kim et al. Page 2 of 3 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 equilibrium value as defined by the fast subsystem and approximate (o()) in Eq. 23 by its period-averaged value () 2π (o (θ; )) dθ 2π = + 2π e bo (θ;) dθ 2π. (25) 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 is approximated by the losed ( s, ) system of equations d s dτ = () s, (26) = q h( s )I(t)g () q 2. (27) with () evaluated in Eq. 25. By self-onsistently solving Eqs. 26 and 27, we an estimate trajetories of the full model when they are near the -nullline in the 2D ( 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 q (Fig. 2) and the slow s -nullline (defined by the relation s = () relating s to ) in terms of. However, the -nullline is plotted in the (q, )-plane while the s -nullline is defined in the (, s )-plane. To plot the nulllines together, we relate the equilibrium value of s, (), to the q oordinate through the monotoni relationship q( s ) = q Ih( () ) = q I( e k () ) and transform the s variable into the q parameter so that both nulllines an be plotted together in the (q, )-plane. These transformed s -nulllines will be denoted q-nulllines. We assume a fixed basal stress input I = and plot the q-nulllines in Fig. 5A 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 q-nullline in (q, )-spae depend strongly on k. Fig. 5B shows a fast -nullline and a slow q-nullline (transformed s -nullline) interseting at both stable branhes of the fast -nullline. Here, the flow field indiates that the 2D projeted trajetory is governed by fast flow over most of the (q, )-spae. 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 on parameters (q,..., p 6, b, k, n, µ, t d ). Other parameters suh as q, q, and µ appear diretly in the fast equation for and thus most strongly ontrol the fast -nullline. Fig. 6A 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. As expeted, numerial simulations of our full model show

13 Kim et al. Page 3 of 3 4 A 4 B 3 inreasing k 3 N 2 2 D q q Figure 5: Slow and fast nulllines and overall flow field. (A) The nullline of s in the ε limit is defined by s = (). To plot these slow nulllines together with the fast -nulllines, we transform the variable s and represent it by q through the relation q = q h( s). These transformed nulllines then beome a funtion of and an be plotted together with the fast -nulllines. For eah fixed value of, o(t; ) is omputed by employing a built-in DDE solver dde23 in MATLAB. The numerial solution is then used to approximate () in Eq. 25 by Euler s method. The q-nullline shifts to the right and gets steeper as k inreases. (B) The fast -nullline defined by qg () = q 2 (blak urve) is plotted together with the slow s-nullline plotted in the (q, ) plane ( q-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. the fast variables (a, o, r) quikly reahing their osillating states defined by the - nullline while the slow variable q = q Ih( s ) remains fairly onstant. Independent of initial onfigurations that are not near the -nullline in (q, )-spae, trajetories quikly jump to one of the stable branhes of the -nullline with little motion towards the q-nullline, as indiated by ξ f in Fig. 6A. One near the -nullline, say when F( s, x) ε, the trajetories vary slowly aording to Eqs. 23. Here, the slow variable s relaxes to its steady state value while satisfying the onstraint F( s, x). In (q, ) spae, the system slowly slides along the -nullline towards the q-nullline (the ξ s paths in Fig. 6A). This latter phase of the evolution ontinues until the system reahes an intersetion of the two nulllines, indiated by the filled dot, at whih the redued subsystem in s and reahes equilibrium. For ertain values of k and if the fast variable is bistable, the two nulllines may interset within eah of the two stable branhes of the -nullline and yield the two distint stable solutions shown in Fig. 6A. For large k, the two nulllines may only interset on one stable branh of the -nullline as shown in Fig. 6B. Trajetories that start within the basin of attration of the lower stable branh of the -nullline ( initial state 2 in Fig. 6B) 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 the full model an be desribed in terms of the loations of the intersetions of nulllines of the redued system.

14 Kim et al. Page 4 of q-nullline -nullline trajetory initial state initial state 2 ξ f ξ s ξ s q ξ f initial state A q-nullline -nullline trajetory initial state 2 ξ f ξ s q ξ s ξ f B initial state Figure 6: Equilibria at the intersetions of nulllines. (A) For intermediate values of k, there are three intersetions, two of them representing stable equilibria. 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 q-nullline (blue) and -nullline (blak). (B) 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 unique intersetion (blak dot) on the upper branh. The senario shown here orresponds to a Type I nullline struture as desribed in the Additional File. Results and Disussion The dual-nullline struture and existene of multiple states disussed above results from the separation of slow CRH synthesis proess and fast CRH seretion proess. This natural physiologial separation of time sales ultimately gives rise to slow dynamis along the fast -nullline during stress. The extent of this slow dynamis will ultimately determine whether a transition between stable states an be indued by stress. In this setion, we explore how external stress-driven transitions mediated by the fast-slow negative feedbak depend on system parameters. 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 (smaller k), the slow q-nullline moves to the left, driving a bistable or fully resistant organism into a stable diseased state. Interventions that inrease k would need to overome hysteresis in order to restore normal HPA funtion. More permanent hanges in parameters are likely to be aused by physial rather than by psyhologial traumas sine suh hanges would imply altered physiology and biohemistry of the person. Traumati brain injury (TBI) is an example of where parameters an be hanged permanently by physial trauma. The injury may derease the sensitivity of the pituitary to ortisol-gr omplex, whih an be desribed by dereasing p 2 in our model. Suh hange in parameter would lead to a leftward shift of the q-nullline and an inreased likelihood of hypoortisolism. In the remainders of this work, we fous on how external stress inputs an by themselves indue stable but reversible transitions in HPA dynamis without hanges in physiologial parameters. Speifially, we onsider only temporary hanges in I(t)

15 Kim et al. Page 5 of 3 and onsider the time-autonomous problem. Sine the majority of neural iruits that projet to the PVN are exitatory [44], we assume external stress stimulates CRH neurons to release CRH above its unit basal rate and that I(t) = + I ext (t) (I base = ) with I ext. To be more onrete in our analysis, we now hoose our nulllines by speifying parameter values. We estimate the values of many of the dimensionless parameters by using values from previous studies, as listed in Table S in the Additional File. Of the four remaining parameters, µ, q, q, and k, we will study how our model depends on k while fixing µ, q, and q. Three possible nullline onfigurations arise aording to the values of µ, q, and q and are delineated in the Additional File. We have also impliitly onsidered only parameter regimes that yield osillations in the PA subsystem at the stable states defined by the nullline intersetions. Given these onsiderations, we heneforth hose µ =.6, q =.4, and q = 77.8 for the rest of our analysis. This hoie of parameters is motivated in the Additional File and orresponds to a so-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. 6A), and one intersetion on the upper stable branh of the -nullline if k > k R (Fig. 6B). For our hosen set of parameters and a basal stress input I =, the ritial values k L = 2.5 < k R = 2.54 are given by Eq. A3 in the Additional File. Normal stress response Ativation of the HPA axis by aute stress ulminates in an inreased seretion of all three main hormones of the HPA axis. Persistent hyperseretion may lead to numerous metaboli, affetive, and psyhoti dysfuntions [45, 46]. Therefore, reovery after stress-indued perturbation is essential to normal HPA funtion. We explore the stability of the HPA axis by initiating the system in the upper of the two stable points shown in Fig. 7A and then imposing a 2min external stress input I ext =.. The HPA axis responds with an inrease in the peak level of ortisol before relaxing bak to its original state after the stress is terminated (Fig. 7B). This transient proess is depited in the projeted (q, )-spae in Fig. 7A. Upon turning on stress, the lumped parameter q and the slow nullline shift to the right by % sine q = q (+I ext )h( () ) (see Fig. 7A). The trajetory will then move rapidly upward towards the new value of on the -nullline; afterwards, it moves very slowly along the -nullline towards the shifted q-nullline. After 2min, the system arrives at the on the -nullline (Fig. 7A). One the stress is shut off the q-nullline returns to its original position defined by I =. The trajetory also jumps bak horizontally to near the initial q value and subsequently quikly returns to the original upper-branh stable point. External stress indues transition from normal to diseased state We now disuss how transitions from a normal to a diseased state an be indued by positive (exitatory) external stress of suffiient duration. In Fig. 8, we start the system in the normal high- state. Upon stimulation of the CRH neurons through I ext >, both CRH and average gluoortioid levels are inreased while the average value of (o(t)) is dereased

16 Kim et al. Page 6 of 3 4 A q 3 B stress, I ext =. 3 2 q o(t) 2 stress end q T (min) Figure 7: Normal stress response. Numerial solution for the response to a 2min external stress I ext =.. (A) At the moment the external stress is turned on, the value of (q, ) inreases from its initial stable solution at (64.4, 27) to (7, 27) after whih the irulating CRH level, quikly reahes the fast -nullline (blak) before slowly evolving along it towards the slow q-nullline (blue). After short durations of stress, the system returns to its starting point within the normal state basin. (B) The peaks of the ortisol level are inreased during stress (red) but return to their original osillating values after the stress is turned off. sine (o) is a dereasing funtion of o. As s (τ) slowly deays towards the dereased target value of (o()), h( s (τ)), and hene q( s ), also derease. As shown in Fig. 8A, muh of this derease ours along the high- stable branh of the - nullline. One the external stress is swithed off, q will jump bak down by a fator of /( + I ext ). If the net derease in q is suffiient to bring it below the bifuration value q L 64 at the leftmost point of the upper knee, 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 q, whih inludes the derease in q inurred during the slow relaxation phase, plus the drop in q assoiated with essation of stress. The minimum duration required for normal-to-diseased transition should also depend on the magnitude of I ext. The relation between the stressor magnitude and duration will be illustrated in the Additional Files. A numerial solution of our model with a 3hr I ext =.2 was performed, and the trajetory in (q, )-spae is shown in Fig. 8A. The orresponding ortisol level along this trajetory is plotted in Fig. 8B, showing that indeed a stable transition to the lower ortisol state ourred shortly after the essation of stress. In addition to a long-term external stress, the stable transition to a diseased state requires 2.5 < k < 2.54 and the existene of two stable points. On the other hand, when k > k R = 2.54, 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 HPA 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

17 Kim et al. Page 7 of stress I ext = o(t) 2 4 A stress end 3 stress I ext =.2 B 3 2 C stress end q o(t) 2 D T (min) Figure 8: Stress-indued transitions into an osillating low-ortisol diseased state. An exitatory external stress I ext =.2 is applied for 3hrs. Here, the system reahes the new stable point set by I =.2 before stress is terminated and the q- nullline reverts to its original position set by I =. (A) At intermediate values of 2.5 < k < 2.54, 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. (B) Numerial solutions of ortisol level o(t ) 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 = 2.54, only the normal stable state exists. The system will reover and return to its original healthy state after a transient period of low ortisol. to evolve along the lower branh towards the q-nullline, eventually sliding off the lower branh near the right bifuration point (indiated in Fig. S2 by (q R, R )) and returning to the single normal equilibrium state. Thus, when k is suffiiently high, the system may experiene a transient period of lowered ortisol level after hroni stress but will eventually reover and return to the normal ortisol state. The orresponding ortisol level shown in Fig. 8D shows this reovery at T 34min, whih ours approximately 5min after the essation of stress. 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. To illustrate this

18 Kim et al. Page 8 of 3 o(t) 3 2 A stress, I ext =. B stress, I ext = T (min) 4 35 C T (min) D stress end q stress end q Figure 9: Stress timing and transition to low-ortisol osillating state. Cortisol levels in response to I ext =. applied over 25min. (A) If stress is initiated at T = 5min, a transition to the low-ortisol diseased state is triggered. (B) If stress is initiated at T = 2min, the system returns to its normal high-ortisol state. Note that the first peak (marked by ) during the stress in (A) is higher than the first peak in (B). (C) If stress is initiated at T = 5min, stress essation and the slow relaxation along the -nullline during stress are suffiient to bring q just left of the separatrix, induing the transition. (D) For initiation time T = 2min, q remains to the right of the separatrix, preluding the transition. dependene on phase, we plot in Figs. 9A and B two solutions for o(t ) obtained with a 25min 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. 9A,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. 9B,D). In this ase, a longer stress duration would be required to push the trajetory past the low-q separatrix into the diseased state. As disussed earlier, an inrease in period-averaged ortisol level during stress drives a normal-to-diseased state transition. We see that the period-averaged level of ortisol under inreased stress is different for stress started at 2min from stress

19 Kim et al. Page 9 of 3 started at 5min. As detailed in the Additional File, the amplitude of the first ortisol peak after the start of stress is signifiantly lower when the applied stress is started 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 onsequently determining whether or not a transition ours. Note that this phase dependene is appreiable only when stress duration is near the threshold value that brings the system lose to the separatrix between normal and diseased basins of attration. Trajetories that pass near separatries are sensitive to small hanges in the overall negative feedbak of ortisol on CRH synthesis, whih depend on the start time of the stress signal. Stress of intermediate duration an indue reverse transitions We an now use our theory to study how positive stressors I ext may be used to indue reverse transitions from the diseased to the normal state. Understanding these reverse transitions may be very 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 we an desribe ET as external stress (I ext > ) applied to a system in the stable low- diseased state. The resulting horizontal shift in q auses the system to move rightward aross the separatrix and suggests a transition to the high- normal state an our upon termination of stress. As shown in Fig. A, if stressor of suffiient duration is applied, the trajetory reahes a point above the unstable branh of the -nullline upon termination leading to the normal, high-ortisol state (Fig. B). 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 q is experiened along the upper stable branh. Cessation of stress might then lower q bak into the basin of attration of the low-ortisol diseased state (Fig. C). Fig. D shows the ortisol level transiently inreasing to a normal level before reverting bak to low levels after approximately 4min. Within our dynamial model, stresses need to be of intermediate duration in order to indue a stable transition from the diseased to the normal state. The ourrene of a reverse transition may also depend on the phase (relative to the intrinsi osillations of the fast PA 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 may be tuned to drive the dynamis of the HPA axis to a normal state in patients with hypoortisolism-assoiated stress disorders. Summary and Conlusions We developed a theory of HPA dynamis that inludes stored CRH, irulating CRH, ACTH, ortisol and gluoortioid reeptor. Our model inorporates a fast self-upregulation of CRH release, a slow negative feedbak effet of ortisol on CRH

20 Kim et al. Page 2 of 3 4 A 3 stress, I ext =.2 B 3 2 o(t) 2 4 C stress end 3 stress I ext =.2 D 3 2 o(t) 2 stress end q T (min) Figure : Stress-indued transitions to high-ortisol osillating state. (A) Projeted 2D system dynamis when a stressor of amplitude I ext =. is applied for 9min starting at T = 2min. is inreased just above the unstable branh ( 2) to allow the unstressed system to ross the separatrix and transition to the normal high- stable state. (B) The plot of o(t ) shows the transition to the high-ortisol, highosillation amplitude state shortly after the 9min stress. (C) A stressor turned off after 78min (3hrs) leaves the system in the basin of attration of the diseased state. (D) Cortisol levels are pushed up but after about 4min relax bak to levels of the original diseased state. synthesis, and a delay in ACTH-ativated ortisol synthesis. These ingredients allow our model to be separated into slow and fast omponents and projeted on a 2D subspae for analysis. Depending on physiologial parameter values, there may exist zero, one, or two stable simultaneous solutions of both fast and slow variables. For small k, CRH release is weak and only the low-crh equilibrium point arises; an individual with suh k is trapped in the low-ortisol diseased state. For large k, only the high-crh normal state arises, rendering the individual resistant to aquiring the long-term, low-ortisol side-effet of ertain stress disorders. When only one stable solution arises, HPA dysregulation must depend on hanges in parameters resulting from permanent physiologial modifiations due to e.g., aging, physial trauma, or stress itself [46, 47]. For example, it has been observed that older rats exhibit inreased CRH seretion while maintaining normal levels of CRH mrna in the PVN [48]. Suh a hange ould be interpreted as an age-dependent inrease in k, whih, in