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Mathematical Models in Neural and Neuroendocrine Systems Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University What’s the Point? Integrate biological data Writing down equations often identifies holes in biological knowledge Use of math models helps to determine if data are logically consistent Making testable predictions Models force simplification, which can help in the identification of essential elements How Much Detail Should Go Into a Model? Some reasons that simpler is better: More detail means more undetermined parameters More equations decrease our ability to comprehend the behavior of the model Added detail can be misleading. It can give the false impression that all the components are necessary for the behavior How Should One Calibrate a Biological Model? In engineering, one does experiments to determine the values of each parameter In biological models it is rare to be able to measure all or most of the parameters… Calibration Method for Biological Models Make model predictions Test them in the lab Modify model Single Cell Models Example: Neural Excitability Hodgkin-Huxley Model (1952) dV Conservation Cm I Na I K I leak I ap of charge dt Gating variables dn n (V ) n / n (V ) dt dm m (V ) m / m (V ) dt dh h (V ) h / h (V ) dt Hodgkin-Huxley Model (1952) Strengths dV Cm I Na I K I leak I ap dt Accurate Essential elements Nothing extra dn n (V ) n / n (V ) dt dm m (V ) m / m (V ) dt dh h (V ) h / h (V ) dt Hodgkin-Huxley Model (1952) Strengths dV Cm I Na I K I leak I ap dt Accurate Essential elements Nothing extra dn n (V ) n / n (V ) dt dm m (V ) m / m (V ) dt dh h (V ) h / h (V ) dt Difficulties Nonlinear High dimensionality Morris-Lecar Model (1981) dV Cm ( I Ca I K I leak I ap ) dt dw w (V ) w / w (V ) dt The m ODE is removed by assuming that the Ca2+ channels activate instantaneously. The h ODE is removed since the Ca2+ channels don’t inactivate. The resulting system is two-dimensional or planar. This greatly simplifies the analysis. How do we analyze this? Phase plane: Study the dynamics in the Vw-plane rather than V or w versus time Nullclines: Determine the curves along which one of the time derivatives is 0 Steady states: At the intersections of the two nullclines both derivatives are 0, so the system is at rest Direction arrows: The nullclines divide up the plane, and the direction of flow in each region can be determined Morris-Lecar with Iap=0 Morris-Lecar with Iap=0 Morris-Lecar with Iap=100 pA Increasing the applied current translates the V-nullcline upward, so that the nullcline intersection is on the middle branch of the V-nullcline. The steady state is unstable. The stable solution is a limit cycle. Morris-Lecar with Iap=100 pA Motion along the limit cycle is periodic. Hopf Bifurcation This transition from a stable steady state to a stable limit cycle through variation of a parameter is called a Hopf bifurcation. It is one way in which periodic motion can arise from a previously stationary system, or vice versa. Morris-Lecar with Iap=250 pA Increasing the applied current more puts the intersection on the right branch of the V-nullcline. Here the steady state is again stable. The system has gone through a second Hopf bifurcation. Bifurcation Diagrams These different behaviors and how they change with variation of a parameter can be summarized with a Bifurcation Diagram. Stationary Branches For each value of the bifurcation parameter Iap plot the V value of the steady state solution. If stable make the curve solid; if unstable make the curve dashed. Stability changes at bifurcation points. Periodic Branches Next plot the maximum and minimum V values of periodic solutions. Stability changes at two Saddle Node of Periodics (SNP) bifurcations. Bistability For some values of Iap the system is bistable. For a single value of the parameter, a stable steady state and a stable limit cycle coexist. Bistability in the Phase Plane The unstable limit cycle (dashed) separates the basin of attraction of the steady state from that of the stable limit cycle. It is the separatrix. Time Courses of Bistable System Example 2: Calcium Dynamics Because Ca2+ plays a large role in cells, modelers have used differential equations to describe the changes over time of the concentration of free Ca2+ in the cytosol, endoplasmic reticulum, and in some cases in the mitochondria. We will derive equations for the free concentration in the cytosol and the ER, starting with the cytosol. The Ins and Outs of Calcium Jin =influx of calcium Jout =efflux of calcium Cac = free calcium concentration d Cac f c J in J out dt fc =fraction of calcium ions not bound to buffer proteins Calcium Flux Terms Influx of calcium is through ion channels, with current ICa: J in I Ca converts current to flux Efflux of calcium is through calcium pumps in the plasma membrane. We can approximate this pumping as proportional to the amount of free calcium in the cytosol: J out k pmcaCac kpmca is the pump rate Altogether Now… d Cac f c I Ca k pmca Cac dt This is just an expression of conservation of mass Calcium Equations with an ER d Cac f c J in J PMCA J leak J IP 3 J SERCA dt d Ca ER f ER J SERCA J leak J IP 3 dt Vc V ER ER Fluxes Jleak is calcium leakage from the ER into the cytosol. This is driven by the calcium concentration difference, so: J leak kleak (CaER Cac ) JSERCA is calcium pumping from the cytosol into the ER through SERCA pumps. For simplicity, we assume that the pump flux is proportional to the level of cytosolic calcium: J SERCA kSERCACac JIP3 is flux through IP3 channels, when activated. Typical Time Courses Mean Field Models What is Mean Field? These models are often used to describe the time dynamics of a large population of cells, where single-cell models would be inappropriate. With a mean field model a single variable would be used to describe, for example, the mean firing rate of a population of neurons. Using single-cell models here would result in thousands of variables and equations. Example: Circadian Prolactin Rhythm Daily PRL rhythm occurs for first 10 days of pregnancy in rats. What is the Mechanism for This? Our hypothesis is that the negative feedback of DA from the arcuate nucleus onto lactotrophs in the pituitary together with the delayed positive feedback of PRL onto DA neurons serves as a rhythm generator. This feedback loop provides delayed negative feedback to lactotrophs. The Dopamine Equation d DA Td k p PRL2 q DA dt DA is mean firing rate of DA neurons PRL is the mean PRL secretory activity Td is tonic stimulatory drive kpPRL2 is the delayed positive feedback of prolactin -qDA is first-order recovery The Prolactin Equation rp d PRL q PRL 2 dt 1 DA The first term has DA in the denominator, reflecting the inhibitory influence of DA on PRL secretion. The PRL-Oxytocin Feedback Loop We and others have shown that Oxytocin (OT) stimulates lactotrophs, and that PRL inhibits OT neurons. Unlike the stimulatory action of PRL on DA neurons, the inhibitory action of PRL on OT neurons is rapid. Modified Prolactin Equation d PRL rp oOT q PRL 2 dt 1 DA The first term now has OT in the numerator, reflecting its stimulatory effect on PRL. The Oxytocin Equation To d OT q OT 2 dt 1 ko PRL The first term has PRL in the denominator, since PRL inhibits OT neurons. There is no time delay. Daily Pulse of VIP VIPergic neurons from the suprachiasmatic nucleus (SCN) innervate DA neurons of the arcuate nucleus and provide inhibition. The VIP activity is high in the morning, and low other times throughout the day. Modified Dopamine Equation d DA Td k p PRL2 q DA rvVIP DA dt The last term, which is negative, reflects the inhibitory action of VIP. The VIP is non-zero for 3 hours in the early morning. VIP Model Reproduces the PRL Rhythm First 6 days Day 3 Uses of the Model We have used the PRL rhythm model to help in the design of experiments. Several predictions have been made and tested, and based on the results we have modified the model. It is a work in progress! Our goal with this mean field model is not to make quantitative predictions, but to make qualitative predictions. This is all that one can expect when thousands of cells are represented by a single variable! Thanks to … Marc Freeman Marcel Egli The NIH and NSF for financial support