The physiological system modelled is delineated in a simple physiological diagram and flow chart designed to show the relationship between cortisol and hippocampal function (Fig and ). SBML was then used to tie together the physiological variables in the model using biochemistry based mathematical questions relating to enzyme substrate reactions (Michaelis-Menten) and receptor ligand interactions (Hill).
The HPA-axis is one of the most studied biological systems and based on available knowledge, a large number of mathematical models of the HPA-axis have been generated [25
]. Such models have focused on recreating the circadian and ultradian rhythms associated with cortisol secretion over 24 hour oscillations. In this paper we were interested in modelling the effects of elevated cortisol levels on hippocampal function over a forty year period. As people age, while it recognised that the circadian rhythms of cortisol secretion are flattened with altered phase and amplitude in older people [30
]. In older people with chronic stress disorders, such as post traumatic stress disorders, and in older people with cognitive impairment, cortisol levels are found to be elevated above the normal diurnal or circadian levels [32
]. Thus the natural rhythms in cortisol secretion were not focussed on in this first model, rather the impact of age, negative feedback and stress were the main features of the model.
There were two principal model outcomes which we were interested in and which relate to clinically described end points in patients with elevated cortisol levels. The first parameter was hippocampal volume (HV)- indexed to hippocampal atrophy, which was a gross measure of decreased neuronal density and dendritic arborisation. The second parameter as hippocampal output (HO); defined as the combined interaction of CA1 excitatory, inhibitory and aging related signals. The CA1 neuron layer is the major output neuronal subfield in the hippocampus. This parameter does not represent hippocampal output in general; rather a simple measure of its temporal activity particularly in relation to AD and declarative memory.
The full list of model variables is included in Table , as are the initial parameter concentrations, and sources used to inform these concentrations, where available. The rate determinants used to define the flow between one variable and another are described in Table and the initial values of the rate constants are detailed in Table . The mathematical expressions used to translate the physiological interactions into in silicoequations are further described. In short hippocampal synaptic activity was described using simple input (u) and output (v) terms for the synaptic current (I)[1
], involving excitatory (Ue) and inhibitory (Ui) synapses and aging related changes in neuronal density, arborisation and growth factors.
Parameter changes to simulate stress and ageing
Model Species and Initial Values
Model Kinetics and Rate Constant Values.
1 HPA Regulation of Cortisol Secretion
The secretion of cortisol via HPA regulation was assumed to be in a "steady-state" and represented using a standard "Michaelis-Menten" type biochemical equations and rate equations (Eq 1–3).
2 Cortisol's Interaction with MR and GR Receptors
The binding of cortisol to CA1 MR receptors was represented using the Hill equation for ligand/receptor binding. The first half of the ordinary differential equation (ODE) represents the generation of MR activity, while the second half of the ODE is a degradation reaction, which is necessary to ensure that MR activity does not rise indefinitely (Eq 4).
The binding of cortisol to GR CA1 hippocampal neurons was expressed also using the Hill equation, and combined with a degradation reaction similar to Eq 4, to ensure that GR receptor activity does not rise indefinitely (Eq 5).
3 Aging of CA1 neurons and cortisol stimulation
The population of neurons in the CA1 region of the hippocampus was defined by combining the stimulation of the neuronal population by neuronal growth factors and MR activity, with the last part of the equation representing the ageing related death of neurons (Eq 6).
A rate constant for the decline of neuronal growth factors with time was also included (Eq 7).
4 Excitatory Input Synapses
Exitatory input synapses (Ue) were defined with reference to the relationship between the neuronal population and excitatory impulses with degradation of neuronal arborisation (Eq 8).
It was assumed that the numbers of excitatory synapses would be related both to the numbers of neuronal branches (dendrites) and also synaptic excitatory signals (Eq 9).
5 Inhibitory Input Synapses
Inhibitory synapses (Ui) were defined by combining the activation of GR receptors with the degradation of synaptic inhibitory signals (Eq 10).
6 Aging related changes in Synaptic Current [Is]
Synaptic current was defined as the net combination of excitatory and inhibitory input synapses (Eq 11).
7 Synaptic Output
Synaptic output (Vs) was defined as a combination of synaptic output with a decline of synaptic output related to time (Eq 12).
8 Hippocampal Atrophy
In the model hippocampus atrophy was defined using an SBML rule which was implemented in MathSBML as detailed below.
9 Hippocampal Tissue Output
The first half of the equation represented the generation of hippocampus output (HO), while the second half of the equation represented the degradation of hippocampus output. The degradation of HO had no biological significance, but served to ensure that hippocampus output did not rise indefinitely when simulations were conducted but instead reached a steady state (Eq 14).
Firstly the system model was brought into a steady state, and the initial hippocampus output was set at 100% which is a mathematical representation of activity, and does not refer to the cognitive ability of the in silicoindividual modelled in this paper.
The response to stress was examined by using events in SBML designed to mirror physiological responses to stress. The first event triggered an increase in the reaction rate kcrh, which raised the secretion of CRH. This in turn precipitated an increase in ACTH, followed by an increase in plasma cortisol. The second event returned kcrh to its original value after a short period of time which produced a yearly increase in cortisol. kcr was not returned to its original value after each event in order so as to represent stress and ageing altering the ability of the HPA-axis to recover from repeated challenges which reflects the clinical hypothesis that ageing impairs homeostatic adaptations of cortisol secretion to stress.(31).