Modeling of oscillations in endocrine networks with feedback

Pattern Formation in a Spatially Extended Model of Pacemaker Dynamics in Smooth Muscle Cells

Spatiotemporal patterns are common in biological systems. For electrically coupled cells, previous studies of pattern formation have mainly used applied current as the primary bifurcation parameter. The purpose of this paper is to show that applied current is not needed to generate spatiotemporal patterns for smooth muscle cells. The patterns can be generated solely by external mechanical stimulation (transmural pressure). To do this we study a reaction-diffusion system involving the Morris-Lecar equations and observe a wide range of spatiotemporal patterns for different values of the model parameters. Some aspects of these patterns are explained via a bifurcation analysis of the system without coupling - in particular Type I and Type II excitability both occur. We show the patterns are not due to a Turing instability and that the spatially extended model exhibits spatiotemporal chaos. We also use travelling wave coordinates to analyse travelling waves.

Modeling pulsativity in the hypothalamic-pituitary-adrenal hormonal axis

A new mathematical model for biological rhythms in the hypothalamic–pituitary–adrenal (HPA) axis is proposed. This model takes the form of a system of impulsive time-delay differential equations which include pulsatile release of adrenocorticotropin (ACTH) by the pituitary gland and a time delay for the release of glucocorticoid hormones by the adrenal gland. Numerical simulations demonstrate that the model’s response to periodic and circadian inputs from the hypothalamus are consistent with those generated by recent models which do not include a pulsatile pituitary. In contrast the oscillatory phenomena generated by the impulsive delay equation mode occur even if the time delay is zero. The observation that the time delay merely introduces a small phase shift suggesting that the effects of the adrenal gland are “downstream” to the origin of pulsativity. In addition, the model accounts for the occurrence of ultradian oscillations in an isolated pituitary gland. These observations suggest that principles of pulse modulated control, familiar to control engineers, may have an increasing role to play in understanding the HPA axis.

An integrate-and-fire model for pulsatility in the neuroendocrine system

A model for pulsatility in neuroendocrine regulation is proposed which combines Goodwin-type feedback control with impulsive input from neurons located in the hypothalamus. The impulsive neural input is modeled using an integrate-and-fire mechanism; namely, inputs are generated only when the membrane potential crosses a threshold, after which it is reset to baseline. The resultant model takes the form of a functional-differential equation with continuous and impulsive components. Despite the impulsive nature of the inputs, realistic hormone profiles are generated, including ultradian and circadian rhythms, pulsatile secretory patterns, and even chaotic dynamics.

Bifurcation phenomena in an impulsive model of non-basal testosterone regulation

Complex nonlinear dynamics in a recent mathematical model of non-basal testosterone regulation are investigated. In agreement with biological evidence, the pulsatile (non-basal) secretion of testosterone is modeled by frequency and amplitude modulated feedback. It is shown that, in addition to already known periodic motions with one and two pulses in the least period of a closed-loop system solution, cycles of higher periodicity and chaos are present in the model in hand. The broad range of exhibited dynamic behaviors makes the model highly promising in model-based signal processing of hormone data.

Association of Basal hyperglucagonemia with impaired glucagon counterregulation in type 1 diabetes

Glucagon counterregulation (GCR) protects against hypoglycemia, but is impaired in type 1 diabetes (T1DM). A model-based analysis of in vivo animal data predicts that the GCR defects are linked to basal hyperglucagonemia. To test this hypothesis we studied the relationship between basal glucagon (BasG) and the GCR response to hypoglycemia in 29 hyperinsulinemic clamps in T1DM patients. Glucose levels were stabilized in euglycemia and then steadily lowered to 50 mg/dL. Glucagon was measured before induction of hypoglycemia and at 10 min intervals after glucose reached levels below 70 mg/dL. GCR was assessed by CumG, the cumulative glucagon levels above basal; MaxG, the maximum glucagon response; and RIG, the relative increase in glucagon over basal. Analysis of the results was performed with our mathematical model of GCR. The model describes interactions between islet peptides and glucose, reproduces the normal GCR axis and its impairment in diabetes. It was used to identify a control mechanism consistent with the observed link between BasG and GCR. Analysis of the clinical data showed that higher BasG was associated with lower GCR response. In particular, CumG and RIG correlated negatively with BasG (r = −0.46, p = 0.012 and r = −0.74, p < 0.0001 respectively) and MaxG increased linearly with BasG at a rate less than unity (p < 0.001). Consistent with these results was a model of GCR in which the secretion of glucagon has two components. The first is under (auto) feedback control and drives a pulsatile GCR and the second is feedback independent (basal secretion) and its increase suppresses the GCR. Our simulations showed that this model explains the observed relationships between BasG and GCR during a three-fold simulated increase in BasG. Our findings support the hypothesis that basal hyperglucagonemia contributes to the GCR impairment in T1DM and show that the predictive power of our GCR animal model applies to human pathophysiology in T1DM.

Optimizing reduction in basal hyperglucagonaemia to repair defective glucagon counterregulation in insulin deficiency

In health, the pancreatic islet cells work as a network with highly co-ordinated signals over time to balance glycaemia within a narrow range. In type 1 diabetes (T1DM), with autoimmune destruction of the β-cells, lack of insulin is considered the primary abnormality and is the primary therapy target. However, replacing insulin alone does not achieve adequate glucose control and recent studies have focused on controlling the endogenous glucagon release as well. In T1DM, glucagon secretion is disordered but not absolutely deficient; it may be excessive postprandially yet it is characteristically insufficient and delayed in response to hypoglycaemia. We review our system-level analysis of the pancreatic endocrine network mechanisms of glucagon counterregulation (GCR) and their dysregulation in T1DM and focus on possible use of α-cell inhibitors (ACIs) to manipulate the glucagon axis to repair the defective GCR. Our results indicate that the GCR abnormalities are of 'network origin'. The lack of β-cell signalling is the primary deficiency that contributes to two separate network abnormalities: (i) absence of a β-cell switch-off trigger and (ii) increased intraislet basal glucagon. A strategy to repair these abnormalities with ACI is proposed, which could achieve better control of glycaemia with reduced hypoglycaemia risk.

Modeling fatigue over sleep deprivation, circadian rhythm, and caffeine with a minimal performance inhibitor model

Sleep loss, as well as concomitant fatigue and risk, is ubiquitous in today's fast-paced society. A biomathematical model that succeeds in describing performance during extended wakefulness would have practical utility in operational environments and could help elucidate the physiological basis of sleep loss effects. Eighteen subjects (14 males, 4 females; age 25.8 +/- 4.3 years) with low levels of habitual caffeine consumption (<300 mg/day) participated. On night 1, subjects slept for 8 h (2300-0700 h), followed by 77 h of continuous wakefulness. They were assigned randomly to receive placebo or caffeine (200 mg, i.e., two sticks of Stay Alert gum) at 0100, 0300, 0500, and 0700 during nights 2, 3, and 4. The psychomotor vigilance test (PVT) was administered periodically over the 77-h period of continuous wakefulness. Statistical analysis reveals lognormality in each PVT, allowing for closed-form median calculation. An iterative parameter estimation algorithm, which takes advantage of MatLab's (R2007a) least-squares nonlinear regression, is used to estimate model parameters from subjects' PVT medians over time awake. In the model, daily periodicity is accounted for with a four-component Fourier series, and a simplified binding function describes asymptotic fatigue. The model highlights patterns in data that suggest (1) the presence of a performance inhibitor that increases and saturates over the period of continuous wakefulness, (2) competitive inhibition of this inhibitor by caffeine, (3) the persistence of an internally driven circadian rhythm of alertness, and (4) a multiplicative relationship between circadian rhythm and performance inhibition. The present inhibitor-based minimal model describes performance data in a manner consistent with known biochemical processes.

Pancreatic network control of glucagon secretion and counterregulation

Glucagon counterregulation (GCR) is a key protection against hypoglycemia compromised in insulinopenic diabetes by an unknown mechanism. In this work, we present an interdisciplinary approach to the analysis of the GCR control mechanisms. Our results indicate that a pancreatic network which unifies a few explicit interactions between the major islet peptides and blood glucose (BG) can replicate the normal GCR axis and explain its impairment in diabetes. A key and novel component of this network is an alpha-cell auto-feedback, which drives glucagon pulsatility and mediates triggering of pulsatile GCR by hypoglycemia via a switch-off of the beta-cell suppression of the alpha-cells. We have performed simulations based on our models of the endocrine pancreas which explain the in vivo GCR response to hypoglycemia of the normal pancreas and the enhancement of defective pulsatile GCR in beta-cell deficiency by switch-off of intrapancreatic alpha-cell suppressing signals. The models also predicted that reduced insulin secretion decreases and delays the GCR. In conclusion, based on experimental data we have developed and validated a model of the normal GCR control mechanisms and their dysregulation in insulin deficient diabetes. One advantage of this construct is that all model components are clinically measurable, thereby permitting its transfer, validation, and application to the study of the GCR abnormalities of the human endocrine pancreas in vivo.

Biomathematical modeling of pulsatile hormone secretion: a historical perspective

Shortly after the recognition of the profound physiological significance of the pulsatile nature of hormone secretion, computer-based modeling techniques were introduced for the identification and characterization of such pulses. Whereas these earlier approaches defined perturbations in hormone concentration-time series, deconvolution procedures were subsequently employed to separate such pulses into their secretion event and clearance components. Stochastic differential equation modeling was also used to define basal and pulsatile hormone secretion. To assess the regulation of individual components within a hormone network, a method that quantitated approximate entropy within hormone concentration-times series was described. To define relationships within coupled hormone systems, methods including cross-correlation and cross-approximate entropy were utilized. To address some of the inherent limitations of these methods, modeling techniques with which to appraise the strength of feedback signaling between and among hormone-secreting components of a network have been developed. Techniques such as dynamic modeling have been utilized to reconstruct dose-response interactions between hormones within coupled systems. A logical extension of these advances will require the development of mathematical methods with which to approximate endocrine networks exhibiting multiple feedback interactions and subsequently reconstruct their parameters based on experimental data for the purpose of testing regulatory hypotheses and estimating alterations in hormone release control mechanisms.

System-level control to optimize glucagon counterregulation by switch-off of α-cell suppressing signals in β-cell deficiency

BACKGROUND

Glucagon counterregulation (GCR) is a key protection against hypoglycemia that is compromised in diabetes. In β-cell-deficient rats, GCR pulsatility can be amplified if insulin (INS) or somatostatin (SS) are infused in the pancreatic artery and then switched off during hypoglycemia. The data indicate that these signals act by different mechanisms, and here we analyze the differences between the two switch offs (SOs) and predict the GCR-amplifying effect of their individual or combined application.

METHODS

A minimal control network (MCN) of α/δ-cell interactions is approximated by differential equations to explain the GCR response to a SO and test in silico the hypotheses: (i) INS SO suppresses basal and pulsatile, while SS SO blocks only pulsatile glucagon release and (ii) simultaneous application of the two switch offs will augment the individual GCR response.

RESULTS

The mechanism postulated in (i) explains the differences in the GCR responses between the SOs. The MCN predicts that simultaneous application of INS and SS decreases basal glucagon but increases post-SO amplitude, thus doubling the response of GCR achieved by each of the individual signals.

CONCLUSION

The current analyses predict that INS and SS SOs improve defective GCR in β-cell deficiency through different but complementary mechanisms and suggest SO strategies to maximally enhance GCR in type 1 diabetes by simultaneous manipulation of the network control. These results are clinically relevant, as they could have application to design of an artificial pancreas by providing ways to augment GCR that would not require glucagon infusion.

Amplification of pulsatile glucagon counterregulation by switch-off of alpha-cell-suppressing signals in streptozotocin-treated rats

Glucagon counterregulation (GCR) is a key protection against hypoglycemia that is compromised in diabetes via an unknown mechanism. To test the hypothesis that alpha-cell-inhibiting signals that are switched off during hypoglycemia amplify GCR, we studied streptozotocin (STZ)-treated male Wistar rats and estimated the effect on GCR of intrapancreatic infusion and termination during hypoglycemia of saline, insulin, and somatostatin. Times 10 min before and 45 min after the switch-off were analyzed. Insulin and somatostatin, but not saline, switch-off significantly increased the glucagon levels (P = 0.03), and the fold increases relative to baseline were significantly higher (P < 0.05) in the insulin and somatostatin groups vs. the saline group. The peak concentrations were also higher in the insulin (368 pg/ml) and somatostatin (228 pg/ml) groups vs. the saline (114 pg/ml) group (P < 0.05). GCR was pulsatile in most animals, indicating a feedback regulation. After the switch-off, the number of secretory events and the total pulsatile production were lower in the saline group vs. the insulin and somatostatin groups (P < 0.05), indicating enhancement of glucagon pulsatile activity by insulin and somatostatin compared with saline. Network modeling analysis demonstrates that reciprocal interactions between alpha- and delta-cells can explain the amplification by interpreting the GCR as a rebound response to the switch-off. The model justifies experimental designs to further study the intrapancreatic network in relation to the switch-off phenomenon. The results of this proof-of-concept interdisciplinary study support the hypothesis that GCR develops as a rebound pulsatile response of the intrapancreatic endocrine feedback network to switch-off of alpha-cell-inhibiting islet signals.

Evidence for acyl-ghrelin modulation of growth hormone release in the fed state

CONTEXT

The timing and frequency of GH secretory episodes is regulated by GHRH and somatostatin. This study provides evidence for amplification of these GH pulses by endogenous acyl-ghrelin.

DESIGN

Blood was sampled every 10 min for 26.5 h during a fed admission with standardized meals and also during the final 24 h of a 61.5-h fast. GH secretion profiles were derived from deconvolution of 10-min sampling data, and full-length acyl-ghrelin levels were measured using a newly developed two-site sandwich assay.

SETTING

The study was conducted at a university hospital general clinical research center.

PARTICIPANTS

Participants included eight men with mean (+/- sd) age 24.5 +/- 3.7 yr (body mass index 24 +/- 2.1 kg/m(2)).

RESULTS

Correlations were computed between amplitudes of individual GH secretory events and the average acyl-ghrelin concentration in the 60-min interval preceding each GH burst. In the fed state, the peak correlations were positive for all subjects and significantly higher than in the fasting state when acyl-ghrelin levels declined [mean (+/- sem): 0.7 (0.04) vs. 0.29 (0.08), P = 0.017]. In addition, long-term fasting was associated with an increase in the GH secretory pulse mass and amplitude but not frequency [fed vs. fasting pulse mass: 0.22 (0.05) vs. 0.44 (0.06) microg/liter, P = 0.002; amplitude: 5.2 (1.3) vs. 11.8 (1.9) microg/liter/min, P = 0.034; pulses per 24 h: 19.4 (0.5) vs. 22.0 (1.4), P = 0.1].

CONCLUSION

Our data support the hypothesis that under normal conditions in subjects given regular meals endogenous acyl-ghrelin acts to increase the amplitude of GH pulses.