Kim et al. RESEARCH Onset, timing, and exposure therapy of stress disorders: mehanisti insight from a mathematial model of osillating neuroendorine dynamis Lae Kim, Maria D Orsogna 2 and Tom Chou 3* * 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 an be haraterized by 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 key physiologial features suh as ultradian osillations, CRH self-upregulation of CRH release, our model distinguishes two omponents of negative feedbak by ortisol on irulating CRH levels: a slow diret suppression of CRH synthesis and afastindireteffetoncrhrelease.cruially,wefindthatthe 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 depressive 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 2 3 4 5 6 7 8 9 0 2 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 group of interating 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) ofthe 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
Kim et al. Page 2 of 28 3 4 5 6 7 8 9 20 2 (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 []. A C 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)Negativefeedbakof 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. 22 23 24 25 26 27 28 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 disrup-
Kim et al. Page 3 of 28 29 30 3 32 33 34 35 36 37 38 39 40 4 42 43 44 45 46 47 48 49 50 5 52 53 54 55 56 57 58 59 60 6 62 63 64 65 66 67 68 69 70 7 72 73 74 tion 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]. 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. Here, hanges in ortisol levels are assumed to arise from anatomial hanges in the HPA axis. 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 assumesthatthe 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, as we shall see, play an essential role in determining the response of the HPA axis to external stress inputs. In this study, we will demonstrate the funtional importane of a number of distintive physiologial features of the HPA axis that have not been previously onsidered in mathematial models. These inlude the effets of intrinsi ultradian osillations on HPA dysregulation, distint fast and slow feedbak mehanisms, 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 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, ACTH, and ortisol are all sereted episodially, with the pulses of ACTH slightly preeding those of ortisol [8]. 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 [9]. 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 shownto influene the physiologial response eliited by the stressor [20]. A seond property
Kim et al. Page 4 of 28 75 76 77 78 79 80 8 82 83 84 85 86 87 88 89 90 9 92 93 is the separation between the synthesis and release proess of CRH whih must be synthesized before being released by CRH neurons in the PVN (Fig. C).CRH release is mediated by synapti signals while synthesis requires the up-regulation of transription and translation. Therefore, the two fundamentally different proesses of release and synthesis operate over very different timesales [2]. 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. 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. A 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 [22], 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 [23]. Models Models of HPA dynamis [3, 4, 6, 7, 24] 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) 05 94 where C(T ),A(T), and O(T ) denote the plasma onentrations of CRH, ACTH, 95 and ortisol, respetively. R(T ) represents the availability of gluoortioid reep- 96 tor (GR) in the anterior pituitary. Cortisol and ortisol-gr omplex are typially 97 near equilibrium so its onentration is approximately proportional to theprod- 98 ut O(T )R(T ) [7]. The parameters p α (α {C, A, O, R}) relate the prodution 99 rate of eah speies to speifi fators that regulate the rate of release/synthesis 00 of the orresponding speies α. External stresses that drive CRH release by the 0 PVN in the hypothalamus are represented by the input signal I(T ). The funtion 02 f C (O) desribes the negative feedbak of ortisol on CRH levels in the PVN while 03 f A (OR, O) desribes the negative feedbak of ortisol or ortisol-gr omplex (at 04 onentration O(T )R(T )) in the pituitary. They are mathematially haraterized as being positive, dereasing funtions so that f A,C ( ) 0andf A,C ( ) < 0. On the other hand, the funtion g R (OR) desribes the self-upregulation effet of the 06 07 08 09 0 ortisol-gr omplex on GR prodution in the anterior pituitary [25]. In ontrast to f A,C ( ), g R ( ) is a positive but inreasing funtion of OR so that g R ( ) 0and g R ( ) > 0. Finally, the degradation funtions d α( ) desribe how eah hormone and reeptor is leared and may be linear or nonlinear.
Kim et al. Page 5 of 28 2 3 4 5 6 7 8 9 20 2 22 23 24 25 26 27 28 29 30 3 32 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, 26]. If f A,C ( ) are Hill-type feedbak funtions dependent only on O(T )andd 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 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 intrinsi iradian [6] or ultradian [3] osillations. However, both models permit only one osillating yle for any given set of parameter values [3, 6] preluding the mathematial distintion between normal and diseased states. Here, we develop a new model of the HPA axis by first introduing a physiologiallymotivated delay diretly into Eq. 3. This delay ultimately gives rise to the observed ultradian osillations [3]. 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, selfupregulation 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. 33 34 35 36 37 38 39 40 Ultradian rhythm and time delay Experiments on rats show a 3-6 minute inherent delay in the responseoftheadrenal gland to ACTH [27]. Moreover, in experiments performed on sheep [28], 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) 4 42 43 44 45 46 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. 47 48 49 50 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
Kim et al. Page 6 of 28 5 52 53 54 55 56 57 58 59 60 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 dt = C (O) C s T C. (6) 6 62 63 64 65 66 67 68 69 70 7 72 73 74 75 76 77 78 79 80 8 82 83 84 85 86 87 88 89 90 9 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 ortisoldependent prodution rate). Eq. 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 [29, 30]. Therefore, we estimate T C 2hrs = 720min. The negative feedbak of ortisol on CRH levels thus atsthrough 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 that had their adrenal glands surgially removed and in whih gluoortioid levels were subsequently kept fixed (by injeting exogenous gluoortioid) for 5-7 days [2, 3]. The measured CRH mrna levels in the PVN were found to derease exponentially with the level of administered gluoortioid [2, 3]. 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. The seretoryresponseof CRH neurons has been shownnot to be diretly sensitive to gluoortioids under many types of stressors, inluding restraint and irulatory shok (hypovolemia) [32]. We limit our attention to these types of stressors and assume that the release of CRH from 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 [33], 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 [34, 35]. 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.. 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 an also be fatored into Eq. through a soure term h(c s ) whih desribes the amount of CRH released per unit of ation
Kim et al. Page 7 of 28 92 93 94 potential ativity of CRH neurons (for example, I(t) desribing the overall firing rate of CRH neurons). Combining the two proposed self-upregulation proesses, we an rewrite Eq. by replaing f C (O) with h(c s )g C (C): dc dt = p CI(T )h(c s )g C (C) d C C. (7) 95 96 97 98 99 In this model, 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. 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. 200 20 202 203 204 205 206 207 208 209 20 2 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. 7); Our omplete mathematial model thus onsists of Eqs. 2, 4, 5, 6, and 7. 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) andg 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, (8) T C dc dt =p CI(T )h(c s )g C (C) d C C, (9) ( ) da dt =p K A AC d A A, (0) K A + OR do dt =p OA(T T d ) d O O, () ( dr dt =p µ R K 2 ) R R KR 2 d R R. (2) +(OR)2 22 23 24 25 26 27 28 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 =0. 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.
Kim et al. Page 8 of 28 29 220 22 222 Nondimensionalization To simplify the further development and analysis of our model, we nondimensionalize Eqs. 8-2 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 (3) d dt = q 0I(t)h( s )g () q 2, (4) da dt = +p 2 (or) p 3a, (5) do dt = a(t t d) o, (6) dr dt = (or)2 p 4 +(or) 2 + p 5 p 6 a, (7) 223 224 225 226 227 228 229 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 (o) = + e bo, h( s )= e ks, (8) µ g () = +(q ) n. 230 23 232 233 234 235 236 237 238 239 240 24 242 243 244 The form of (o) is based on the above-mentioned exponential relation observed adrenaletomized rats [2, 3]. 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. 3. The redued prodution of s will then lead to a smaller h( s ) and ultimately a redued release soure for (Eq. 4). 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. 2 and 7. 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.
Kim et al. Page 9 of 28 245 246 247 248 249 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, andaslow timesale τ = t/t εt. The full model (Eqs. 3-7) an be suintly written in the form d s dt = ε( (o) s ), (9) dx dt = F( s, x), (20) 250 25 252 253 254 255 256 where x =(, a, o, r) is the vetor of fast dynamial variables, and F( s, x) denotes the right-hand-sides of Eqs. 4-7. We refer to the fast dynamis desribed by dx/dt = F( s, x) asafast flow. Intheε 0 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. 20 (Eq. 4) anbe written as d dt = q( s(τ),i)g () q 2, (2) 257 258 259 260 26 262 263 264 265 266 267 268 269 270 27 272 273 274 275 276 277 278 279 280 where q( s (τ),i) q 0 Ih( s (τ)) = q 0 I( e ks(τ ) ) is a funtion of s (τ) andi. Sine s is a funtion only of the slow timesale τ, q an be viewed as a bifuration parameter ontrolling, over short timesales, the fast flow desribed by Eq. 2. One (t) quikly reahes its non-osillating quasi-equilibrium value defined by d/dt = qg () q 2 = 0, it an be viewed as a parametri term in Eq. 5 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,andq, 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. 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 0 limit, irulating CRH only feeds forward into a, o, andr, a omplete desription of all the fast variables an be onstruted from just whih obeys Eq. 2. 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.
Kim et al. Page 0 of 28 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. 2 an be visualized as the intersetion of the two funtions qg () (dashedurve)andq 2 (gray line). For a given Hill-type funtion g (), Eq. 2 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. 2) with q as the bifuration parameter. Solid and dashed segments represent stable and unstable steady states of the fast variables, respetively. LandUlabelbasinsofattrationassoiatedwiththelowerand upper stable branhes of the -nullline. Left and right bifuration points (q L, L)and(q R, R)areindiated. Fixed points of appear and disappear through saddle node bifurations as q is varied through q L and q R. 3 a * () a * () A 3 o* () o *() B 2 2 physiologial regime 0 0 0 20 30 40 50 physiologial regime 0 0 0 20 30 40 50 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 [36] and t d =.44, orresponding to a delay of T d =5min. 28 282 To analyze the evolution of the slow variable s (τ), we write our equations in terms of τ = εt:
Kim et al. Page of 28 slow variable fast variables x(t) ( ) τ in (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) arefastvariables.ofthese,(a, o, r) formthetypiallyosillatory PA-subsystem that is reapitulated by. Intheε =/t limit,thevariable s(τ) slowly relaxes towards a period-averaged value (o()). Therefore,thefullmodel an be aurately desribed by its projetion onto the 2D ( s,)phasespae. d s dτ =( (o) s ), (22) ε dx dτ = F( s, x). (23) 283 284 285 286 287 288 In the ε 0 limit, the outer solution F( s, x) 0 simply onstrains the system to be on the fast -nullline defined by qg () =q 2. 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 equilibrium value as defined by the fast subsystem and approximate (o()) in Eq. 22 by its period-averaged value () 2π 0 (o (θ; )) dθ 2π = + 2π 0 e bo (θ;) dθ 2π. (24) 289 290 29 292 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, (25) 0=q 0 h( s )I(t)g () q 2. (26) 293 294 295 296 297 298 with () evaluated in Eq. 24. By self-onsistently solving Eqs. 25 and 26, 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
Kim et al. Page 2 of 28 299 300 30 302 303 304 305 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, (), totheq oordinate through the monotoni relationship q( s )=q 0 Ih( () ) =q 0 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. 40 A 40 B 30 inreasing k 30 N 20 20 D 0 62 64 66 68 q 0 62 64 66 68 q Figure 5: Slow and fast nulllines and overall flow field. (A) The nullline of s in the ε 0limitisdefinedby s = ().Toplottheseslownulllinestogetherwiththe fast -nulllines, we transform the variable s and represent it by q through the relation q = q 0h( 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; ) isomputed by employing a built-in DDE solver dde23 in MATLAB. The numerial solution is then used to approximate () in Eq. 24 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. 306 307 308 309 30 3 32 33 34 35 36 37 38 39 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 ofourmodel in the small ε limit. In general, the number of allowable nullline intersetions will depend on input level I and on parameters (q 0,..., p 6,b,k,n,µ,t d ). Other parameters suh as q 0, q,andµ 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
Kim et al. Page 3 of 28 320 32 322 323 324 325 326 327 328 329 330 33 332 333 334 335 336 337 338 339 340 the fast variables (a, o, r) quikly reahing their osillating states defined by the - nullline while the slow variable q = q 0 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. 22. Here, the slow variable s relaxes to its steady state value while satisfying the onstraint F( s, x) 0. 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. 40 35 30 25 20 q-nullline -nullline trajetory initial state initial state 2 ξ f ξ s ξ s 5 62 66 70 q ξ f initial state A 40 35 30 25 20 q-nullline -nullline trajetory initial state 2 ξ f ξ s 5 62 66 70 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 trajetoriesapproah 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.
Kim et al. Page 4 of 28 34 342 343 344 345 346 347 348 349 350 35 352 353 354 355 356 357 358 359 360 36 362 363 364 365 366 367 368 369 370 37 372 373 374 375 376 377 378 379 380 38 382 383 384 385 Results and Disussion The dual-nullline struture and existene of multiple states disussed above results from the separation of the negative feedbak on CRH into slow and fast proesses. This natural physiologial separation of time sales that ultimately gives rise to slow dynamis along the fast -nullline. The extend of these dynamis will ultimately determine whether a transition between stable states an our. 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 permanent diseased state. Interventions that inrease k would need to overome hysteresis in order to restore normal HPA funtion. More permanent physiologial hanges suh as those resulting from traumati brain injury (TBI) may derease the sensitivity of the pituitary to ortisol-gr omplex. This hange would be desribed by dereasing p 2 in our model, leading to a leftward shift of the q-nullline and an inreased likelihood of hypoortisolism. In this study, we fous on how external stress inputs an by themselves indue permanent but reversible transitions in HPA dynamis without hanges in physiologial 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 [37], 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 0. 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 0,q,andk, we will study how our model depends on k while fixing µ,q 0,andq. Three possible nullline onfigurations arise aording to the values of µ,q 0,andq 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 =0.6, q =0.04, and q 0 =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 [38, 39]. Therefore, reovery after stress-indued perturbation is essential to normal HPA funtion. We
Kim et al. Page 5 of 28 386 387 388 389 390 39 392 393 394 395 396 397 398 explore the stability of the HPA axis by initiating the system in the upper ofthetwo stable points shown in Fig. 7A and then imposing a 20min external stress input I ext =0.. 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 0% sine q = q 0 ( + 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 20min, the system arrives at the onthe-nullline. One the stress is shut off the q-nullline returns to its original position defined by I =.Thetrajetory also jumps bak to near (q, ) =(64.4, 36) and subsequently quikly returns to the original upper-branh stable point. 40 A q 3 B stress, I ext =0. 30 20 q o(t ) 2 stress end 0 62 65 68 7 q 0 0 240 480 720 T (min) Figure 7: Normal stress response. Numerial solution for the response to a 20min external stress I ext =0.. (A) At the moment the external stress is turned on, the value of (q, ) inreasesfromitsinitialstablesolutionat(64.4, 27) to (7, 27) after whih the irulating CRH level, quiklyreahesthefast-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) butreturntotheiroriginal osillating values after the stress is turned off. 399 400 40 402 403 404 405 406 407 408 409 External stress indues transition from normal to diseased state We now disuss how permanent 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 > 0, 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. 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
Kim et al. Page 6 of 28 40 4 42 43 44 45 46 47 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. A numerial solution of our model with a 30hr I ext =0.2 wasperformed,andthe 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 permanent transition to the lower ortisol state ourred shortly after the essation of stress. In addition 40 3 stress I ext =0.2 30 20 o(t ) 2 0 40 A stress end 0 3 stress I ext =0.2 B 30 20 C stress end 0 62 66 70 74 78 q o(t ) 2 D 0 0 440 2880 4320 T (min) Figure 8: Stress-indued transitions into an osillating low-ortisol diseased state. An exitatory external stress I ext =0.2 isappliedfor30hrs.here,thesystem reahes the new stable point set by I =.2 beforestressisterminatedandtheqnullline 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 )plottedagainsttheoriginaltimevariablet 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. 48 49 420 42 to a long-term external stress, the permanent 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
Kim et al. Page 7 of 28 422 423 424 425 426 427 428 429 430 43 432 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 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 3400min, whih ours approximately 500min after the essation of stress. o(t ) 3 2 A stress, I ext =0. B stress, I ext =0. 0 0 480 960 440 T (min) 40 35 C 0 480 960 440 T (min) D 30 25 20 stress end 5 62 65 68 7 q stress end 62 65 68 7 q Figure 9: Stress timing and transition to low-ortisol osillating state. Cortisol levels in response to I ext =0. appliedover250min.(a)ifstressisinitiatedatt = 50min, a transition to the low-ortisol diseased state is triggered. (B) If stress is initiated at T =20min,thesystemreturnstoitsnormalhigh-ortisolstate. 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 =50min,stressessationandtheslowrelaxation along the -nullline during stress are suffiient to bring q just left of the separatrix, induing the transition. (D) For initiation time T =20min,q remains to the right of the separatrix, preluding the transition.
Kim et al. Page 8 of 28 433 434 435 436 437 438 439 440 44 442 443 444 445 446 447 448 449 450 45 452 453 454 455 456 457 458 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 dependene on phase, we plot in Figs. 9A and B two solutions for o(t ) obtained with a 250min I ext =0.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 = 000min (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 20min from stress started at 50min. 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. 459 460 46 462 463 464 465 466 467 468 469 470 47 472 473 474 475 476 477 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 > 0) 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. As shown in Fig. 0A, if the stressor is applied long enough, the trajetory reahes a point above the unstable branh of the -nullline upon termination leading to the normal, high-ortisol state (Fig. 0B). 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. 0C). Fig. 0D shows the ortisol level transiently inreasing to a normal level before reverting bak to low levels after approximately 400min.
Kim et al. Page 9 of 28 40 A 3 stress, I ext =0.2 B 30 20 o(t ) 2 0 40 C stress end 0 3 stress I ext =0.2 D 30 20 o(t ) 2 stress end 0 62 66 70 74 q 0 0 640 280 920 T (min) Figure 0: Stress-indued transitions to high-ortisol osillating state. (A) Projeted 2D system dynamis when a stressor of amplitude I ext =0. isappliedfor 9min starting at T =20min. is inreased just above the unstable branh ( 20) to allow the unstressed system to ross the separatrix and transition to the normal high- stable state. (B) The plot of o(t )showsthetransitiontothehigh-ortisol,highosillation amplitude state shortly after the 9min stress. (C) A stressor turned off after 780min (3hrs) leaves the system in the basin of attration of the diseased state.(d) Cortisol levels are pushed up but after about 400min relax bak to levels of the original diseased state. 478 479 480 48 482 483 484 485 486 487 488 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 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. 0C. 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.
Kim et al. Page 20 of 28 489 490 49 492 493 494 495 496 497 498 499 500 50 502 503 504 505 506 507 508 509 50 5 52 53 54 55 56 57 58 59 520 52 522 523 524 525 526 527 528 529 530 53 532 533 534 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 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, ortwo stable simultaneous solutions of both fast and slow variables. For small k, CRH release is weak and only the low-crh equilibrium point arises; suh an individual is trapped in the low-ortisol diseased state. For large k, only the high-crh normal state arises, rendering the organism 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 [39, 40]. For example, it has been observed that older rats exhibit inreased CRH seretion while maintaining normal levels of CRH mrna 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 [42, 43]. 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 HPA 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 HPA funtion, possibly deoupling stress disorders from hypoortisolism. Interestingly, we show that the minimum durations required 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 negative-feedbak effet on CRH an determine whetherornot