Articles in PresS. Am J Physiol Regul Integr Comp Physiol (February 5, 2014). doi:10.1152/ajpregu.00365.2013

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Meth Math: Modeling Temperature Responses to Methamphetamine

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Yaroslav I. Molkov,1 Maria V. Zaretskaia,2 and Dmitry V. Zaretsky2,*

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Department of Mathematical Sciences, Indiana University – Purdue University Indianapolis, IN

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Department of Emergency Medicine, Indiana University School of Medicine, Indianapolis, IN

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Running title: Meth Math

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*Corresponding author:

Dmitry V. Zaretsky, M.D., Ph.D.

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Department of Emergency Medicine

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Indiana University School of Medicine

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635 Barnhill Dr, MS 438

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Indianapolis, IN 46202-5120

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E-mail: [email protected]

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1 Copyright © 2014 by the American Physiological Society.

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ABSTRACT

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Methamphetamine (Meth) can evoke extreme hyperthermia, which correlates with both

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neurotoxicity and death in laboratory animals and humans. The objective of this study was to

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uncover the mechanisms of a complex dose-dependence of temperature responses to Meth by

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mathematical modeling the neural circuitry involved. Based on previous studies, we composed

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an artificial neural network with the core comprised of three sequentially connected nodes:

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Excitatory, Medullary and sympathetic preganglionic SPN. Meth directly stimulated the

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Excitatory node, an inhibitory drive targeting the Medullary node, and in high doses - additional

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excitatory drive affecting the SPN. All model parameters (weights of connections, sensitivities,

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time constants) were subject to fitting experimental time series of temperature responses to 1, 3,

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5, and 10 mg/kg of Meth. Modeling suggested that the temperature response to the lowest dose

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of Meth which caused an immediate and short hyperthermia, involves neuronal excitation at a

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supramedullary level. The delay in response seen after intermediate doses of Meth is a result of

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neuronal inhibition at the medullary level. Finally, the rapid and robust increase in body

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temperature induced by the highest dose of Meth involves activation of high-dose excitatory

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drive. The impairment in the inhibitory mechanism can provoke life-threatening temperature

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rise, and makes it a plausible cause of fatal hyperthermia in Meth users. We expect that studying

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putative neuronal sites of Meth action and involved neuromediators resulting in a detailed model

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of this system may lead to more effective strategies of prevention and treatment of hyperthermia

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induced by amphetamine-like stimulants.

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INTRODUCTION Derivatives of amphetamines are widely abused all over the world. After long-term use cognitive, neurophysiological, and neuroanatomical deficits have been reported (10, 11, 47, 62, 66, 69, 75). Neurophysiological deficits are enhanced by hyperthermia (2) which itself is a major mortality factor in drug abusers (27, 74). Despite numerous studies investigating the mechanisms behind methamphetamine-induced hyperthermia there are still no specific treatments. A key barrier in the design of specific treatments is a lack of consensus regarding which brain regions and receptors are involved.

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Methamphetamine (Meth) has a complicated dose-related temperature response. Relatively

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low doses of Meth (≤1 mg/kg) cause a short lived but rapid increase in body temperature (49,

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59). As the dose of Meth increases, dose-response curve is not linear. Rather, at doses between 1

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and 5 mg/kg the peak temperatures do not increase but the temperature peak time progressively

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shifts to the right: for example, the peak temperature for a 1 mg/kg dose (38˚C) occurs at 60 min

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while the peak for a 5 mg/kg dose (38˚C) occurs at ~180 minutes. After doses of Meth above 10

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mg/kg temperature once again rapidly rises, peaking between 60 and 90 min, followed by a small

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decrease and then secondarily rises again with temperature remaining elevated 5 hr after

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injection (49). Importantly, temperature responses to injections of Meth are dependent on ambient temperature and may include both hypothermic and hyperthermic phases (38). Similar complex temperature responses have been reported with other amphetamines including the substituted amphetamine MDMA (56).

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Collectively these responses point out the complex pharmacology of amphetamines and in particular Meth. Meth increases multiple neurotransmitters including dopamine, norepinephrine,

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acetylcholine, glutamate, and serotonin (32, 53, 72). Meth can also act directly on sigma opioid

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receptors (41), and trace amine receptors (78). Multiple interactions between these

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neurotransmitters and receptors make it difficult to study and understand the mechanisms behind

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Meth’s temperature effects. Further complicating the analysis, body temperature is dependent on

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multiple thermoregulatory mechanisms and complex neuronal circuitry.

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As both acute complications (including fatalities) and chronic neurotoxicity from Meth are

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linked to its effects on temperature (2, 31), a better understanding of how Meth affects body

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temperature may provide insight into prevention and treatment of these effects.

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When administered in low doses, amphetamines in general, and methamphetamine in

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particular, induce an array of responses strikingly similar to those evoked by the stimulation or

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disinhibition of neurons in the region of the dorsomedial hypothalamus (DMH) in conscious rats:

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tachycardia, mild hypertension, hyperthermia through increase of thermogenesis and suppression

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of heat dissipation, and activation of locomotion (DMH: (14, 51, 65, 68, 81); Amphetamines:

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(38, 48, 58)). In turn, inhibition of the DMH with microinjections of muscimol significantly

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attenuated hyperthermia, tachycardia, hypertension and locomotion evoked by a systemic dose of

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MDMA (3,4-methylenedioxymethamphetamine) (58). Structural similarity between MDMA and

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methamphetamine allows us to hypothesize that the DMH may also be involved in mediating

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responses to methamphetamine.

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In the past decade, DMH has emerged as a key hypothalamic effector region whose

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activation plays an important role in generating fever and stress responses (for reviews see (15-

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17, 22, 77)). The DMH exerts most of its autonomic effects through medullary structures: rostral

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ventrolateral medulla (RVLM, (21)) and ventromedial medulla (VMM, (7, 9, 63)). Peripheral

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responses mediated by the DMH, including thermoregulatory ones, are predominantly

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transmitted through the raphe pallidus (RP) in the VMM. Among those effects are non-shivering

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thermogenesis including thermogenesis in the interscapular brown adipose tissue (IBAT, (7, 81);

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shivering, which can result in production of significant amount of heat (6, 39), and control of

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heat dissipation through cutaneous blood flow (43, 44).

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The sources of input to the DMH are multiple (42, 70), however their functional roles and

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relative weights in mediating specific component of thermoregulatory processes remain

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undescribed. Inhibition of the DMH can suppress responses to the activation of amygdala (67)

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and dlPAG (13). Stimulation of amygdaloid region affects cutaneous blood flow (36), which is a

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major part of thermoregulatory process. Are those connections direct and involved in mediation

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of stress, fever, or in responses to drugs of abuse, is still unknown. Therefore, in our study we

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described Meth as acting directly on the DMH, while this could be not the case.

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Our experimental data contained clear evidence of a presence of the inhibitory drive

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(response to intermediate dose is weaker than the one to the low dose). Both the DMH and the

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RP have inhibitory inputs: disinhibition of both areas by antagonists of GABAA receptors evokes

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thermogenic responses (37, 81) and decreasing heat dissipation through the constriction of

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cutaneous vasculature (43). One of main sources of GABA-ergic projections to both areas is

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located in preoptic area, however, the cells projecting to each one are not the same (40, 79).

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Inhibitory tone to the RP could also originate from ventrolateral periaqueductal gray (vlPAG)

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(52) and rostral ventrolateral medulla (RVLM) (8). However, while inhibitory component of

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responses to Meth could be due to activation of inhibitory projection (presynaptic effect), it can

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also be mediated by activation of dopamine receptors (most likely D2, (45, 55)), or alpha2-

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adrenergic receptors (34) located directly on neurons (postsynaptic effect). Without experimental

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support of any specific pathway or receptor involved, for clarity of description we modeled our

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inhibitory component as an inhibitory drive to the RP.

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Finally, (46) demonstrated that 5-HT-2A receptors in the spinal cord contribute to cutaneous

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vasoconstriction after a high dose of MDMA. The high-dose component is clearly present in our

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data: high dose of Meth overrides the inhibition, and body temperature sharply goes up, followed

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by a long plateau of elevated body temperature.

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Previous studies pinpoint a few brain areas possibly involved in mediating the responses to

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Meth, however, in the following description we avoided specific locations as they have not been

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verified experimentally, relying instead on generic descriptors of nodes (Fig. 1). We refer to the

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dorsomedial hypothalamus as the “Excitatory node”, and its projection to the “Medullary node”

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will compete for activation of the latter with the aggregate inhibitory drive, which is coming

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from the “Inhibitory node”. The high-dose component, which cannot be inhibited by the

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inhibitory drive, enters the thermogenic pathway at the inframedullary SPN node.

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Through an iterative process we were able to create various neural networks, each of which

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closely models observed temperature responses from a range of Meth doses. Despite being

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obviously different from each other, those networks share few core principles of organization,

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main of which is a competition between converging excitatory and inhibitory drives.

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METHODS

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Experimental Procedures

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All procedures described were approved by the Indiana University Institutional Animal Care and Use Committee and followed NIH guidelines.

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Animal model

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Male Sprague-Dawley rats (280-350 g) were individually housed with a 12 h light cycle at a

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room temperature of 23-25oC with free access to food and water. All animals for which data are

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reported remained in good health throughout the course of surgical procedures and experimental

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protocols as assessed by appearance, behavior, and maintenance of body weight.

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Surgical Procedures For measurements of core temperature, as well as heart rate and blood pressure, rats were

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implanted with telemetric transmitters (C50-PXT, Data Sciences Int., St.Paul, MN) under

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isoflurane anesthesia as previously described (82). Catheter was inserted into abdominal aorta

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through femoral artery, and the body of the transmitter was placed into the abdominal cavity and

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sutured to the abdominal wall. After at least seven days of recovery, rats, still in their home

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cages, were brought to experimental room. Cages were placed on telemetric receivers (RPC-1),

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and animals were given at least two hours to adapt to the new environment.

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Drugs Methamphetamine hydrochloride was obtained from the Sigma-Aldrich (St. Louis, MO). It was dissolved in sterile saline at the time of injections and injected at a volume of 1 ml/kg.

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Determining Temperature Responses to Methamphetamine

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The temperature response to various doses of methamphetamine was determined by

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randomizing animals receiving an intraperitoneal injection of either saline or one of the four

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doses of Meth (1, 3, 5 or 10 mg/kg). Each animal received only one injection with six rats per

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dose. Data were recorded every two minutes, transferred to Microsoft Excel, and 10 min

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averages were calculated using a template in Excel.

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Statistical analysis The results are presented as the mean ± SEM. Results were compared using a one way

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ANOVA with repeated measures followed by a Fisher’s LSD post hoc test, where appropriate. A

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value of P50 min for the dose of 3 mg/kg and t>90 min for

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the dose of 5 mg/kg) were similar to response to low dose of Meth to the accuracy of a time shift.

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Similar to 5 mg/kg, the highest dose of Meth tested (10 mg/kg) caused an immediate increase

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in temperature. The increase however was greater (~2˚C above baseline) and required a longer

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period of time to peak (~80 min). After reaching a peak, the temperature was falling over the

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next 60 minutes and then, similar to the 5 mg/kg dose, formed a second peak at least in some

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animals, or stayed relatively constant for ~60 min. Unlike 5 mg/kg, second peak was not strongly

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expressed, and temperature was slowly returning to baseline levels ~6 h after injection (data not

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shown).

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Collectively these data show a complex multimodal temperature response to Meth that is highly dependent on the dose.

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Estimated parameters and model validation Data shown in Fig. 2 were used to find the optimal set of parameters of each of the models as

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described in Methods. The calculated parameter values with their standard error estimates are

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listed in Table 1. To evaluate the goodness of fit we calculated the coefficient of determination

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R2 provided in Table 2 for each model and all doses used. This coefficient can be treated as a

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fraction of the variance explained by the models, which ranges between 65% and 95% for

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different doses and on average constitutes approximately 90% for all three models. We also

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calculated the ratio of the root-mean-square of the model residuals to the standard deviations of

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temperature responses over the group of animals (N=6). Table 3 contains these calculations for

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all three models and all doses used. The fact that in all cases this measure is much less than

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100% confirms that the model mismatch is well within experimentally observed animal-to-

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animal variability.

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To validate the model we used k-fold cross validation. Four temperature time series as a

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single data set were randomly divided into 8 sets of points of equal size. Every set out of 8 was

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used for validation, and the remaining 7 sets were used to optimize the model parameters. The

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residuals for the validation sets were collected and root-mean-squared to be compared with the

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root-mean-square residuals for the training sets. The ratios of root-mean-squared residuals for the

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validation and training sets constituted 1.12 for “3 arrows” model, 1.08 for “2 arrows” model and

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1.09 for “1 arrow” model. These ratios can be compared to the expected ones in case of linear

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regression. The size of each training set was n=77 data points. The models had p=13, 10 and 10

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parameters respectively. Accordingly, the expected ratios of root-mean-squared residuals of

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validation and training sets

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numbers obtained are slightly less than the estimates probably due to regularization used in our

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optimization procedure. In general, it evidences the robustness of the models.

/

were approximately 1.18, 1.14 and 1.14. The

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Model simulation and interpretation These parameters were used to simulate the dynamics of Meth blood concentration, activities

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of the neuronal populations and temperature responses to different doses of Meth under various

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(patho)physiological conditions as described below.

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Pharmacokinetics

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The dependence of the Meth blood concentration on time is described by Eq. (2) where the

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time constant for absorption

defines a rate at which Meth is initially accumulated in the blood

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due to its absorption from the peritoneum, and the elimination time constant

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how fast the drug is washing out. After parameter optimization to fit experimental data in all

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models the absorption appeared to be much faster (

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values) than the elimination (

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parameters, after an i.p. injection of Meth, the blood concentration quickly peaks at a maximum

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of about 70% of the dose, and then exponentially decreases with a time constant of about 1 h.

is responsible for

~ 10 min, see Table 1 for model specific

~ 60 min, see Table 1 for model specific values). With such

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Activity in the Meth-sensitive nodes in “3 arrows” model In this model, which was able to accurately reproduce the observed temperature curves for all

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4 doses of Meth (Fig. 3C), there were three neuronal nodes stimulated by Meth directly (Fig.

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3A): Exc, Inhib, and an excitatory projection to the SPN node which was activated only at high

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doses of Meth (HD). The activation function of each Meth-sensitive population depended on two

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parameters: half-activation concentration and the maximal slope. The slopes appeared to be

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comparable for all nodes, while a half-activation concentration was the lowest for Exc, and the

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highest for HD (Fig. 3B and Table 1, “3 arrows” model).

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Activation time courses of each of these nodes for each dose of Meth are shown in Fig. 3C.

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All doses of Meth were able to activate the Exc node which reached its 100% activity at 3 mg/kg

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for a short period of time. Each successively higher dose activated this node for a longer period

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of time. Similarly, the Inhib node was activated by all tested doses of Meth. The sensitivity of

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this node was less than of sensitivity of Exc requiring higher doses of Meth for full activation

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and having shorter durations of maximal activity at the higher doses. Finally, the high dose (HD)

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component was activated only at the highest two doses of Meth tested (5 and 10 mg/kg).

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Temperature responses to low and low intermediate doses The activity of the SPN, which defines an output of the system, can be viewed as a

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competition between excitatory and inhibitory drives. Excitatory drive in 3-arrow model is a sum

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of drives coming from Exc and HD, which is offset by activity of the Inhib. To visualize how

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various doses of Meth result in activity of the SPN, Fig. 4 presents activation function for

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combined excitatory drive and inhibitory drive (Fig. 4A) matched with pharmacokinetic profiles

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of Meth after various doses in this study (Fig. 4C).

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At the lowest dose of Meth Exc was the only activator of SPN in the model (curve 1 in Fig.

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4C), and inhibition from Inhib was not sufficient to suppress excitation. Therefore, the activity of

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the SPN population followed the activity of Exc.

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At 3 mg/kg dose and higher the inhibitory input from Inhib was able to almost fully suppress

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the activity of Mdl population which mediated an activation of SPN. This is why no significant

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SPN activation was observed immediately after the injection of the lower intermediate dose in

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Fig. 3C, except short-lived peak at the rising shoulder of Meth concentration. The activity of

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SPN started rising after Meth blood concentration fell enough to deactivate Inhib, but the

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concentration transiently remained high enough to maintain Exc activity. Accordingly, the

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duration of the latency period to extended activation of the SPN was dose-dependent: the higher

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the dose is, the longer it takes to washout Meth. Thus, our model suggests that during late phases

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of responses, the SPN population exhibits identical activity patterns to the precision of the time

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shift which monotonically increases with dose.

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Temperature responses to high intermediate and high doses The maximal Meth blood concentration after the higher intermediate dose of 5 mg/kg was

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sufficient to slightly activate the HD node (Fig. 4A, third row of traces): appr 3.5 mg/kg at max

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is in the very beginning of the HD step (the third vertical line in Fig. 3B). Accordingly, the

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activity of SPN becomes bimodal: the first maximum was evoked by a slight and immediate HD

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activation, and the secondary maximum was mediated by the mechanism identical to the delayed

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response to 3 mg/kg dose: disinhibition of Mdl from Inhib during Meth washout with still present

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activity of the Exc.

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As the dose increased from 5 to 10 mg/kg, HD component was activated to a much greater

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extent (Fig. 3C). Interestingly, in spite of the fact that the two components of the excitatory drive

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(Exc and HD) have similar magnitudes, the HD activation led to three-fold stronger activity of

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SPN compared to similar activation of Exc, whose effect was significantly attenuated by the

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inhibitory drive activation. At high doses the inhibitory response was saturated, and could not

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counteract excessive thermogenesis.

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Reduced models

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Model schematic shown in Fig. 3A (“3-arrows” model) assumes that there are two distinct

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Meth-dependent excitatory nodes with significantly different sensitivities – Exc and HD – which

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collectively determine the excitatory drive to the SPN node. Together they form a two-step

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response curve shown by a solid line in Fig. 4A. This curve can be treated as an activation curve

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of an aggregate population composed of both nodes. This allows for the reduction of the model

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in such a way that two excitatory Meth-sensitive nodes can be combined into a single node

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characterized by single sigmoid activation curve (Fig. 4B). We call such a reduced model “2-

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arrows” model according to the number of sites affected by Meth (Fig. 5A).

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“2-arrows” model

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By eliminating the HD node from the model schematic, the circuitry can be refigured as

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shown in Fig. 5A to have only two Meth sensitive nodes. In addition, this allows combining the

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Mdl and SPN nodes into single SPN node. With the optimal set of parameters (Table 1) and

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corresponding activation curves for Meth-sensitive nodes (Fig.5B), this two-arrow model also

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accurately reproduces the observed data (Fig. 5C) predicting temperature responses within one

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standard deviation of the data for all times used for the fitting procedure (220 min, gray bars in

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Fig. 5C).

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To place activation functions of Excitatory and Inhibitory nodes into context of effective

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concentrations of Meth, we have plotted those (Fig.4B) together with matching pharmacokinetic

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profiles (Fig.4C). The response of the Exc node to Meth injections does not saturate at the

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intermediate doses, but continues to gradually increase as the dose increases. This results in the

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monotonically increasing curve of the excitatory component (solid line on Fig. 4B and Fig. 5B).

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The Inhib node, as in the three-arrow model, has a sigmoidal activation function (Fig. 5B).

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Unlike in the “3-arrows” model, the activation curves of Exc and Inhib nodes have

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significantly different slopes (Figs. 4B, 5B; Table 1) which results in a different mechanism of

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the Meth dose-dependence. After the lowest dose the excitatory drive provides an increase of the

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SPN activity, while the inhibitory input barely reacts (Fig. 4B,C, line “1”). With increasing dose,

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the Inhib node abruptly activates, the growth of inhibition prevails over steady but moderate

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increase in excitation, and the SPN becomes silent (Fig. 4B,C, line “3”). However, the inhibitory

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drive eventually saturates, while the excitatory input to SPN continues to grow (Fig. 4B,C,

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lines”5” and “10”). That results in activation of the SPN by sufficiently high doses of Meth.

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“1-arrow” model

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In the “2-arrows” model both Exc and Inhib populations are affected by Meth independently.

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However, one can see that the entire range of the Inhib’s sensitivity to Meth lies within the linear

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Exc response (the entire “step” of the dashed sigmoid in Fig. 3B happens while the solid curve is

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gradually increasing). From modeling perspective this means that we can leave only one Meth

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sensitive input in the schematic, and replace individual Inhib sensitivity to Meth by the

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excitatory input from Exc (Figs. 6A). Such schematic is supported by known activatory

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projections from the DMH to the RVLM (80). At optimal parameter values (see Table 1) this “1-arrow” model produces population

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activation patterns indistinguishable from the “2-arrow” model (compare Figs. 5C and 6C), and

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accurately predict what was observed in the experiment.

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Temperature variability in response to parameter variations

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Experimental temperature curves exhibit significant animal-to-animal variability. For

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instance, after injection of 10 mg/kg of Meth the maximal temperature can differ between

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animals by more than 2°C (see standard deviations at Fig. 1). To evoke similar alterations in the

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model responses we perturbed some model parameters relative to their optimal values in Table 1.

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Specifically, we varied synaptic weights of the projections from Exc and Inhib to Mdl in both “2-

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“ and “3-arrows” models, and additionally from HD to SPN in the “3-arrows” model, i.e.

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E

M

,

I

M

and

HD PSN .

The increase in

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E

M

and

HD PSN

with a

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corresponding decrease in E

M

and

HD PSN

I

M

produced the most exaggerated response and a decrease in

with an increase in

I

M

brought an attenuation of response.

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Figure 7 shows how a 3% change of the control parameters relative to their optimal values

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altered the temperature curves produced by the 2-arrows model (A) and 3-arrows model (B). In

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both models this change led to temperature variations of the same order of magnitude as

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experimentally observed variability (compare top and bottom thick solid curves with error bars

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representing experimental standard deviations). Interestingly, such perturbations lead to

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noticeably greater temperature variations for the intermediate and the highest doses in the 2-

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arrows model as compared to the 3-arrows one.

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Significant suppression of the inhibitory drive results in dramatic amplification of

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hyperthermia induced by methamphetamine (Fig. 8). In the absence of inhibition the response to

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1 mg/kg approximately doubles, while a peak of the response to 3 mg/kg exceeds 40 oC, and is

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close to life-threatening levels. With inhibition suppressed by 50%, hyperthermia is not that

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dramatic, however after low doses body temperatures reaches levels typical for responses to high

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doses.

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DISCUSSION

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Meth-evoked hyperthermia can cause death by itself, but also is known to aggravate

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neurological consequences of acute or chronic use of amphetamines. Despite significant efforts

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to uncover reasons why some people develop life-threatening hyperthermia, while most people’s

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thermal response is not a medical catastrophe, we still do not have a clear idea of how

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hyperthermia develops. In this study we attempted to develop an integrative model of

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temperature response to amphetamines in general and to Meth in particular.

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Dose-dependence of the temperature responses to Meth appeared to be not trivial —

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intermediate doses of methamphetamine (Meth, 3 or 5 mg/kg, see Fig.2) caused less

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hyperthermia than both lowest (1 mg/kg) and highest (10 mg/kg) doses of the drug. Also,

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responses to the lowest and the highest doses were virtually instantaneous, while the responses to

509

intermediate doses were delayed. The highest dose evokes profound hyperthermia followed by a

510

secondary peak of temperature. Our results were qualitatively similar to those described in (38).

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Our approach to mathematical description of responses was to model the involved neuronal

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circuitry in the form of an artificial neural network (1). Some of the nodes of the circuitry

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putatively corresponded to actual neuronal populations involved in thermoregulatory responses

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to amphetamines in accordance with current physiological conception. All model parameters

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were chosen so that the model had the best fit to the experimental time series. When we used the

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optimal set of parameters, the simulated values of body temperature were within one SD of

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actual experimental data throughout all time courses for all used doses (see Table 3). Our model

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was additionally validated by comparing the “best-fit” values of some measurable parameters

519

with experimental values available from literature. For example, in all models presented blood

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plasma Meth half-life was estimated to be about 1 h (or more precisely - 57 min); same

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parameter in experimental studies was found to be 40-50 min (23) or appr. 70 min (35).

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More importantly, along with parameters which can be measured in experimental settings

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with relative ease, we reconstructed the responses of the involved neuronal ensembles. Such an

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approach may be a powerful tool for inferring the dynamics of individual functional populations

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since there are limited options for measuring neuronal activity in conscious freely moving

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animals.

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Mechanisms of multimodal responses in “3-arrows” model Activities of the neuronal populations directly stimulated by Meth ultimately converge at the

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SPN population whose activity is treated as the term responsible for thermogenesis in Eq. (6).

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Increases of body temperature induced by Meth are compensated by heat dissipation which is

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proportional to the difference from baseline body temperature: baseline temperature is the one at

533

which non-Meth-induced heat generation and dissipation are at equilibrium. For specific Meth

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blood concentration the activity of SPN, whose patterns are depicted in the fourth row of traces

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at Fig. 3C, is defined by the difference between excitatory and inhibitory components. The

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dependence of the excitatory and inhibitory drives on Meth blood concentrations (Fig.4A) is

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shown against pharmacokinetic profiles for all four doses (Fig. 4C).

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The lowest dose of Meth (1 mg/kg) activates neither the inhibitory node nor HD. Thermal

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output at this dose is dependent on activity of a single multisynaptic pathway from the excitatory

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node through the medulla to the SPN node. Activation of the Exc follows the blood concentration

541

of Meth, so does the body temperature.

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The increase of the dose to 3 mg/kg is insufficient to activate SPN through HD, but it exceeds

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the threshold of activation of inhibitory drive. Output of the Mdl is dependent on a balance

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between excitatory and inhibitory drives: the difference between the dotted line and the dashed

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curve (Fig. 4A) becomes very small to the right from the second vertical line corresponding to 3

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mg/kg dose. Before the inhibitory drive kicks in, there is a short spike of supramedullary

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excitation (Fig.3C) which is successfully transmitted through the Mdl during absorption phase,

548

but this spike is very short and is not sufficient to noticeably change body temperature.

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In experimental settings inhibition of the medullary relay prevents even maximal

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supramedullary (hypothalamic) excitation to be transmitted downstream (7, 9, 44, 63). This quite

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fits our model data: the inhibitory drive successfully competes with the supramedullary

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excitatory drive, so at the lower intermediate dose there is no immediate increase of body

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temperature. Although the absence of an initial reaction may look like a delay in response, it is in

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fact a result of a competition between two responses: excitatory and inhibitory ones.

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Dose-dependence of appearance of excitatory and inhibitory components is of critical

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importance for interpretation of dose-dependence of responses to Meth. In neurocircuitry-based

557

model excitatory and inhibitory components differ in sensitivity to Meth. Most likely such

558

difference is due to predominant receptor mechanism responsible for the component. Meth

559

evokes release of multiple neuromediators with excitatory drive is most likely being dopamine-

560

mediated (5, 28, 61). In turn, inhibitory component could have multiple origins in

561

monoaminergic systems – dopaminergic through D2 receptors (45), adrenergic through alpha2-

562

adrenoreceptors (34) or evenserotonergic through 5-HT1A receptors (55, 57). Inhibition could be

563

actually not direct, but mediated by facilitation of the inhibitory GABA-ergic drive, tonically

564

present in this circuitry (26). Variety of potentially involved mechanisms can explain differences

565

of sensitivity of components of the model to Meth. Also, in our modeling we assumed uniform

566

distribution of Meth, while this assumption is not necessarily valid (33).

567 568

To discover specific mechanisms of physiological processes, modelling approach allows generating testable hypotheses. An assumption, that the inhibitory drive is mediated by dopamine

26

569

or adrenoreceptors, allows predicting the dynamics of the response to a specific dose of Meth

570

under blockade of those receptors (see Fig. 8 for an example).

571

Noteworthy, the intermediate doses result in a sharp and very short-lived peak in the activity

572

of the Mdl/SPN at the rising shoulder of the blood Meth concentration profile before inhibition

573

kicks in (Figs. 3, 5, 6). This activation is so short, that, due to inertia of the thermal system (the

574

temperature time constant

575

Presence of this short-lived initial peak predicted by all models could be the source of

576

misinterpretations in neuroanatomical studies if such a marker of neuronal activity as c-fos

577

immunoreactivity is used. Even though the peak is too short to evoke a significant thermal

578

response, this few minutes long burst still could be sufficient to induce expression of c-fos. Once

579

expressed, c-fos is present in nuclei for few hours (25). In such situation the marker of neuronal

580

activity will be present without clear physiological response.

581

90 min), no significant increase in the body temperature occurs.

The higher of the intermediate doses (5 mg/kg) did not activate HD significantly, but for a

582

short time the Meth level was elevated enough to evoke mild HD response. This drove a short-

583

lived activation of the SPN, which in turn resulted in a slight increase of the body temperature

584

seen in Fig. 2 immediately after the injection. However, the HD-evoked activation does not last

585

long, so temperature does not further increase after initial rise.

586

Due to elimination, after both the lower (3 mg/kg) and higher (5 mg/kg) intermediate doses

587

blood levels of Meth eventually drop below the threshold that is needed to keep the inhibitory

588

drive active. However, these levels are still sufficient to maintain the Exc activity that serves as

589

an excitatory drive for the Mdl. The latter in turn activates SPN which drives hyperthermia. Since

590

the elimination time constant is significantly larger than absorption one (

591

min, Table 1), for times

57 min vs

the dynamics of the Meth blood concentrations are defined

27

8

592

almost exclusively by elimination. By the time when blood concentration of Meth falls to the

593

levels at which the excitatory input to the Mdl overcomes the inhibition, the body temperature

594

starts drawing similar patterns for all low and intermediate doses because pharmacokinetics of

595

Meth following that moment is similar. So the late phases of temperature responses to

596

intermediate doses are virtually identical to the response to a lowest dose (see 1, 3 and 5 mg/kg

597

on Fig. 2).

598

Unlike the low and intermediate doses, the highest dose activates HD significantly. In fact,

599

our model suggests that activation of HD provides remarkably stronger activation of the SPN in

600

comparison with supramedullary component (Fig. 3C, compare SPN activities at different doses).

601

Interestingly, the Exc alone provides input of similar magnitude (see Fig. 4A, the first step on the

602

solid line), but it is significantly attenuated by the inhibitory counterpart (Fig. 4A, dashed line).

603

That is what makes the slope of the temperature curve in the beginning of the response to the

604

highest dose the greatest as compared to other doses.

605 606

Sustained exposure to Meth

607

The suggested model can be used to give an estimate of maximal body temperature which

608

will be reached if thermogenic pathways are activated to the utmost extent for a prolonged period

609

of time due to, for example, repeated Meth administration. The excess of this steady state

610

temperature above the baseline temperature (see Eq. (6)) is defined by the largest possible

611

activity of the SPN node which can be estimated as the difference between saturation levels of its

612

excitatory and inhibitory inputs (see solid and dashed lines on Fig. 4A for high Meth

613

concentrations). This difference is approximately 5oC, which implies a highest possible

28

614

temperature of 42o C. Evidently, high doses of Meth are able to evoke life-threatening

615

hyperthermia, even in conditions of room ambient temperature.

616

Similar estimations can be done for the hyperthermia evoked by low doses. If a low dose of

617

Meth approximately equal to 1 mg/kg is maintained in blood stream for long time (repeated

618

administration) the excess of the steady state temperature above the baseline will constitute ~2oC

619

(see Fig. 4A), which means an absolute body temperature of 39oC. This is not life-threatening by

620

itself, but is definitely outside normal temperature range for healthy animals kept at room

621

temperature.

622

Interestingly, our model predicts the milder or even no hyperthermia for sustained

623

intermediate concentrations of Meth since the excitatory input to SPN can be (completely)

624

suppressed by the inhibitory one for the Meth blood concentrations of 2-3 mg/kg (solid and

625

dashed lines almost coincide on Fig. 4A in this range).

626 627

Simplified model with two inputs for Meth

628

In the original “3-arrows” model the activation of the Meth-sensitive nodes happens in a

629

binary manner (Fig. 4A). They can be thought as triggers switched on and off as the blood Meth

630

concentration crosses their activation thresholds. The arrangement of the thresholds (half-

631

activation concentrations) defines a temporal activation pattern of the network in response to a

632

particular dose (Fig. 3B,C; Table 1). Interestingly, the model predicts comparable slopes of the

633

activation curves for all three Meth-sensitive nodes

634

activation thresholds represent excitability of corresponding neuronal populations which can be

635

controlled by a number of synaptic or intrinsic mechanisms.

29

E

,

I

and

HD

(Fig. 3B; Table 1). The

636

A key feature of the model that defines the absence of thermogenic response when the

637

intermediate doses of Meth are maintained in the blood is that the curve of the inhibitory

638

component activation (dashed line on Fig. 4A) touches the excitatory one (solid line on Fig. 4A).

639

Same feature allows for a different modeling solution implemented in the “2 arrows” model (Fig.

640

4B). In this simplified model we combined the low-dose Exc and the high-dose HD components

641

into a single excitatory drive gradually activated throughout the entire range of Meth variation

642

(Figs. 5A, 4B). We also combined the SPN with the Mdl into a single node of sympathetic

643

premotor neurons (SPN). Unlike in the “3-arrows” model, where all doses of Meth activate the

644

Exc to almost maximal values, in the simplified “2-arrows” model different doses of Meth

645

activate the Exc to a different extent. Increasing the dose not only prolongs stimulation, but also

646

increases the amplitude. At the same time the inhibitory component has an activation curve

647

similar to one in the “3-arrows” model. Accordingly, after high doses, Inhib already saturates,

648

while the activity of Exc continues to grow with dose, and that creates an immediate “high-dose”

649

component of hyperthermia (see lines “5” and “10” on Fig. 4B,C).

650 651

Model with single input for Meth

652

As we have mentioned above, the excitatory and inhibitory inputs in the “2-arrows” model as

653

functions of the blood Meth concentration appear to have significantly different slopes (see Figs.

654

4B and 5B). The inhibitory response curve is much steeper which makes the excitatory one

655

virtually linear in the corresponding range of Meth concentrations. Accordingly, the response of

656

the Inhib population to the changes of Meth in the blood can be formed by the synaptic inputs

657

from the Exc, and not due to its intrinsic sensitivity to Meth (see Fig. 6A). This assumption

658

together with one about actual neuroanatomical prototypes of the nodes of the model (Exc is the

30

659

DMH, Inhib is RVLM) is supported by the existing functional projections from the DMH to the

660

RVLM (21, 80). This “1-arrow” model represents an extreme case when the variety of possible

661

temperature patterns is dictated by a specific network organization, while the system receives a

662

single Meth-dependent input.

663 664

Which model is correct?

665

In this initial approach to develop a mathematical model of temperature responses to

666

amphetamines, we did not intend to create an all-inclusive “ultimate” model of the circuitry

667

involved in those responses. We took this first step to break into potential mechanisms defining

668

non-trivial dose-dependence of responses to amphetamines. We showed that this complex

669

phenomenon can be explained by relatively simple neuronal network architecture comprising a

670

core of the temperature control system.

671

We considered important to present various potential circuitries that may underlie

672

phenomena difficult to explain using qualitative approach typical for pharmacodynamics. In our

673

study, all three models replicated the experimental data with comparable precision. However,

674

each model described herein can be used to generate testable hypotheses for their subsequent

675

experimental verification. One of the most obvious approaches to verify the models may be

676

either inactivation or activation of the putative anatomical structures involved in responses to

677

Meth. For example, we hypothesize that the supramedullar node is the DMH, hence the “3

678

arrows” model implies that inhibition of the DMH should prevent responses to the lowest dose,

679

will not affect initial phase of response to highest dose of Meth, but will suppress the late phase

680

of responses to the intermediate and the highest doses.

681

31

682

Variability of responses and life-threatening hyperthermia

683

Our mathematical models can be used to gain an insight into potential mechanism of life-

684

threatening hyperthermia induced by amphetamines. We found that just 3% change of certain

685

parameters is sufficient to produce significantly modified responses (Fig.7).

686

“High-dose” component is relatively short-lived – less than 60 min after 10 mg/kg: this

687

length defines the maximum of intense hyperthermia. Prolongation of this component would

688

make rising shoulder after high dose longer, and, therefore, body temperature will be able to

689

reach life-threatening levels. Therefore, dramatically higher responses to amphetamines could

690

appear due to purely pharmacokinetic, not pharmacodynamic factors, such as increase of half-life

691

or repeated administration.

692

Importantly, “low dose” is not showing its power only due to prompt activation of inhibitory

693

drive. If not compensated by inhibitory drive, the effect of the excitatory drive after low doses of

694

Meth is comparable with intensity of average “high-dose” component of response. This predicts

695

that if for any reasons the inhibitory drive is not activated by Meth, even low doses of Meth will

696

evoke high-dose-like response, with life-threatening levels of hyperthermia developing after

697

“physiological” doses (Fig. 8). In the extreme situation, when no inhibitory drive is activated in

698

the “3-arrows” model, the calculations show that the dose of 3 mg/kg will result in body

699

temperature of 40.5 in 100 min (see Fig. 8). Catastrophic consequences of inhibitory failure may

700

be a plausible explanation of a wide range of blood levels which can result in fatalities after

701

amphetamine overdose (12): some cases could be due to actual overdose, while some could be

702

due to abnormal response to relatively low doses.

703 704

Missing parts of model / Future directions

32

705

In any of these models we did not include a component concerned with stress evoked by

706

manipulations with an animal. In fact, the disturbance caused by i.p. injection could significantly

707

increase the body temperature of a conscious rat. The amplitude of the neuronal response to

708

injection could be comparable to the amplitude of response to the lowest dose of Meth, while

709

usually it is significantly shorter. Interestingly, slight hyperthermia due to stress of injection does

710

not seem to appear at the intermediate doses (see Fig.1). Pathways, which are involved in the

711

response to both amphetamines and stress, are shared (58). With that in mind, it is quite logical

712

that activation of the inhibitory pathway will suppress both the excitatory drive induced by Meth,

713

and prevent hyperthermia from stress. For simplicity sake we did not include a stress component

714

in these studies, however, this is our plan for future developments.

715

In presented models we used τT as a constant. That implicitly considers that no

716

thermoregulatory changes, such as cutaneous vasodilation, occur in response to an increase in

717

body temperature. In experiments which are performed at room temperature rats normally do not

718

thermoregulate through dissipation of heat. This mechanism only activates when the body

719

temperature increases above a certain threshold (54). Activation of sympathetic system by Meth

720

(59) offsets this control mechanism toward higher thresholds,thus eliminating or at least greatly

721

attenuating the feedback concerned with hyperthermia.

722

Feedback mechanisms are activated when conditions are deviating from thermoneutrality,

723

and, therefore, addition of feedback mechanisms to the model will be critical for proper

724

description of responses at extreme conditions, first of all in hot or cold environments. In those

725

conditions activity of already functional feedback mechanisms could be modified by the drug.

726

For example amphetamine analog MDMA (ecstasy) suppresses thermogenesis induced by cold

727

(55). Fortunately, afferent pathways and feedback mechanisms were extensively modelled

33

728

previously (18-20, 29, 30, 64, 73). However, adding such components to the model requires data

729

obtained in varying environmental conditions, and was beyond the scope of the current study.

730

When developing the model, we assumed that medullary node is the raphe pallidus (RP). It is

731

known that even at room ambient temperature inhibition of the RP results in a profound drop of

732

the body temperature (82) which implies that RP normally exhibits substantial tonic activity. The

733

assimilation of this data will allow introducing additional constraints on the model parameters.

734

It is known that administration of amphetamines results in thermodysregulation: at elevated

735

ambient temperatures the drug results in hyperthermia, while at low ambient temperature animals

736

become hypothermic after administration of the same drug (61). This implies that pathways

737

involved in responses to both amphetamines and changes of ambient temperature are shared.

738

However, inclusion of the ambient temperature as an independent parameter into the model will

739

require formal description of thermoregulatory processes. We consider the above as the next step

740

in constructing a closed loop model of body temperature control with the ultimate goal to explain

741

how Meth modulates/disrupts temperature regulation.

742 743 744

CONCLUSION Our interdisciplinary experimental and modeling study revealed that several relatively simple

745

models are able to describe the complex pharmacodynamics of Meth. Our models had few

746

common features elucidating the essential mechanisms of Meth-evoked hyperthermia. First, the

747

thermal outcome is defined by the interaction of excitatory and inhibitory drives, both of which

748

are activated by Meth. Second, the low-dose induced component of the excitatory drive can be

749

completely suppressed by the inhibitory drive. Inadequate activation of the inhibitory component

750

of response to amphetamines may be the reason of fatal hyperthermia. Third, the high dose of

34

751

Meth activates a component of the excitatory drive, which cannot be compensated by the

752

inhibitory drive, either because of its location or insufficient strength of the inhibitory

753

component.

754 755 756

PERSPECTIVES AND SIGNIFICANCE One of the most favorable outcomes of modeling is a generation of testable hypothesis. The

757

common features of the models described in this manuscript are, in fact, such testable

758

hypotheses. While we developed the model with specific brain areas in mind (such as the DMH,

759

the RVLM, the RP and the spinal cord), the actual representation of nodes remains untested. To

760

test our hypotheses about involved neurocircuitry, the activity of the abovementioned neuronal

761

populations needs to be experimentally targeted. Also, the neuromediators involved in synaptic

762

transmission between these nodes are yet to be determined. We expect that targeted modulation

763

of activity of various brain structures and further pharmacological testing will reveal the real

764

faces of those schematic nodes and arrows. The detailed model may provide a powerful tool for

765

developing new strategies and therapies against amphetamine evoked life-threatening

766

hyperthermia.

767 768 769

Acknowledgements

770

Research reported in this publication was supported by the National Institute on Drug Abuse

771

of the NIH under award number R01DA026867 and iM2CS-GEIRE. Furthermore, this work was

772

conducted in a facility constructed with support from the National Center for Research

773

Resources, of the NIH under award number C06 RR015481-010. Pamela Durant is gratefully

774

acknowledged for editorial assistance. 35

775

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1026

47

1027

FIGURE LEGENDS

1028

Fig. 1. Neuronal circuitry involved in responses to amphetamines. A. Actual anatomical structures. B.

1029

Simplified conceptual network. Lines with arrows – excitatory projections (source is white-colored); lines

1030

with circles – inhibitory projections (source is grey-colored).

1031 1032

Fig. 2. Dose-dependence of temperature responses to Meth. Injections were performed i.p. at t=0 min in a

1033

volume of 1 ml/kg.

1034 1035

Fig.3. “3 arrows” (circuitry-based) model. A. Model schematic. Each circle represents a neural

1036

population. Meth-sensitive populations (marked by arrow with label “Meth”) are modeled as an

1037

artificial neuron with sigmoid activation function applied to its input (see text for a detailed

1038

description). The circuitry compiles the literature data (see Introduction). B. Activation functions

1039

of Meth-sensitive populations. Vertical lines show half-activation concentrations. C.

1040

Reconstructed activity of neuronal populations included into the model together with comparison

1041

of reconstructed dynamics of body temperature after various doses of Meth with actual

1042

experimental data used in fitting procedures. All reconstructed values of body temperature were

1043

within SD of actual experimental data within time period used for fitting procedures (220 min,

1044

grey rectangle).

1045 1046

Fig. 4. A, B. The excitatory (solid line) and inhibitory (dashed line) components of the

1047

thermogenic activity in “3 arrows” (A) and in “2 arrows” (B) models as functions of blood

1048

concentration of Meth. C. Reconstructed time courses of Meth blood concentration for 4 doses of

1049

Meth from 1 to 10 mg/kg. Vertical lines show where the maximal Meth blood concentrations for

1050

each dose fall on the graphs A and B.

48

1051 1052 1053

Fig. 5. “2 arrows” model. See legend for Fig.3.

1054 1055

Fig. 6. “1 arrow” model. See legend for Fig.3.

1056 1057

Fig. 7. Variability of temperature responses to four doses of METH due to model parameter

1058

perturbation. Doses are shown on the top. A. “2 arrows” model. B. “3 arrows” model. Black

1059

filled circles with error bars represent average and standard deviation of experimentally

1060

measured core body temperature. Upper solid lines are the maximal temperature responses, and

1061

lower solid lines are minimal temperature responses produced by the models after 3% change of

1062

key model parameters.

1063 1064

Fig. 8. Modeling temperature responses to Meth (1 and 3 mg/kg) if inhibitory component is

1065

suppressed. Closed circles with error bars – experimental data; thin solid line – best-fitting “3-

1066

arrows” model; dashed line – model with all parameters of thin solid line except the weight of

1067

inhibitory projection is considered 50% of the original value; thick solid line – the weight of

1068

inhibitory projection is considered equal 0.

1069

49

1070 1071 1072 1073 1074

TABLES Table 1. Optimal model parameters with standard errors. Optimization was performed as described in “Model parameter estimation” section in Methods. Model Parameter

3-arrows

2-arrows

1-arrow

(min)

8.25 ± 1.12

11.2 ± 1.1

11.3 ± 1.1

(min)

57.5 ± 1.02

57.2 ± 1.2

57.1 ± 1.2

(min)

89.2 ± 2.65

78.4 ± 2.8

79.2 ± 2.8

E E E

(a.u.)

-0.357 ± 0.006 -0.437 ± 0.002

-0.376 ± 0.005

(mg/kg)-1

1.225 ± 0.015

0.375 ± 0.002

0.337 ± 0.005

/

0.29

1.17

1.12

(mg/kg)*

E

(a.u.)

-1.335 ± 0.013 -1.746 ± 0.008

-4.27 ± 0.01

(mg/kg)-1

1.463 ± 0.017

1.140 ± 0.007

N/A

/

0.91

1.53

N/A

N/A

N/A

7.47 ± 0.02

(a.u.)

-3.69 ± 0.05

N/A

N/A

(mg/kg)-1

0.872 ± 0.013

N/A

N/A

4.24

N/A

N/A

-3.35 ± 0.02

-7.20 ± 0.02

-7.62 ± 0.02

9.89 ± 0.03

N/A

N/A

6.38 ± 0.04

N/A

N/A

I

wI I E

(a.u.)

I

HD HD

(mg/kg)*

I

HD / HD SPN E I

(deg C) M

(deg C) (deg C)

M

wHD

SPN

(deg C)

5.66 ± 0.12

N/A

N/A

E

SPN

(deg C)

N/A

24.86 ± 0.04

23.82 ± 0.04

N/A

12.12 ± 0.05

10.44 ± 0.05

I

1075 1076 1077

(mg/kg)*

SPN

(deg C)

* calculated half-activation Meth concentrations.

50

1078 1079

Table 2. Coefficient of determination R2 as a measure of models goodness of fit calculated as R2 = 1 - Var(residuals)/Var(average temperature). Model / Dose

1 mg/kg

3 mg/kg

5 mg/kg

10 mg/kg

Overall

3 arrows

76.8%

96.0%

65.8%

96.9%

90.7%

2 arrows

66.9%

95.4%

64.4%

94.7%

87.8%

1 arrow

65.9%

94.9%

63.8%

94.8%

87.6%

1080 1081 1082 1083

Table 3. Model root-mean-square residual relative to the standard deviation of the temperature over the group of animals (N=6 for each group) root-mean-squared over time of observation. Model / Dose

1 mg/kg

3 mg/kg

5 mg/kg

10 mg/kg

Overall

3 arrows

41.9%

17.7%

55.7%

17.4%

29.0%

2 arrows

50.0%

19.0%

56.8%

22.9%

33.1%

1 arrow

50.8%

20.1%

57.3%

22.6%

33.5%

1084 1085 1086

51

B

MPOA DMH

RVLM

dlPAG

RP

SC

Shivering thermogenesis



Rostrocaudal distribuon

vlPAG

Inhibitory

A

Excitatory

Medullary

Inframedullary

Heat Accumulaon

)LJ

Meth

A

HD

C

3 mg/kg

5 mg/kg

10 mg/kg

TEMP

SPN

Meth Inhib

Exc (a.u.)

Mdl

0.5

Inhib (a.u.)

Exc

0.5

HD (a.u.)

1

Meth

B

1 mg/kg

0.5

0 1

0 1

0

0.8 0.6 0.4 0.2 0

4

40 39 38 37

2 0

100 min

0

2

4

6 METH (mg/kg)

)LJ

SPN (°C)

Exc Inhib HD

TEMP (°C)

1

8

10

Inputs to SPN (°C)

A

16 14 12 10 8 6 4 2

HD Exc 3 arrows

Excitatory Inhibitory Inputs to SPN (°C)

B

26 22 18

2 and 1 arrows

14 10 6

C

0

Time after i.p. injection (h)

1

3

5

10

1

2

3

4

0

2

4

6

METH blood concentration (mg/kg)

Fig. 4.

8

10

A Meth

Exc

C

TEMP

SPN

1 mg/kg

3 mg/kg

5 mg/kg

10 mg/kg

Exc (a.u.)

1

Inhib (a.u.)

1

Exc Inhib

0.8 0.6 0.4

0 1 0.5 0

SPN (°C)

B

Inhib

4

TEMP (°C)

Meth

0.5

40 39 38 37

2 0

0.2 100 min

0

0

2

4

6 METH (mg/kg)

)LJ

8

10

A METH

Exc

SPN

TEMP

Inhib

B

1 mg/kg

3 mg/kg

5 mg/kg

10 mg/kg

0.5 0 1 0.5

SPN (°C)

0 4

TEMP (°C)

Inhib (a.u.)

Exc (a.u.)

1

40 39 38 37

2 0

100 min

Fig. 6.

TEMP (°C)

A

1 mg/kg

3 mg/kg

5 mg/kg

10 mg/kg

40 39 38 37

TEMP (°C)

B

40 39 38 37 100 min

)LJ

41

1 mg/kg

3 mg/kg no Inhib 50% Inhib

Temperature (ºC)

40

original

39

38

37 

100 min