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Introduction to Mathematical Methods in Neurobiology: Dynamical Systems Oren Shriki 2009 First Order Differential Equations
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Two Types of Dynamical Systems Differential equations: Describe the evolution of systems in continuous time. Difference equations / Iterated maps: Describe the evolution of systems in discrete time.
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What is a Differential Equation? Any equation of the form: For example:
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Order of a Differential Equation The order of a differential equation is the order of the highest derivative in the equation. A differential equation of order n has the form:
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1 st Order Differential Equations A 1 st order differential equation has the form: For example:
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Separable Differential Equations Separable equations have the form: For example:
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Separable Differential Equations How to solve separable equations? If h(y)≠0 we can write: Integrating both sides with respect to x we obtain:
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Separable Differential Equations By substituting: We obtain:
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Example 1
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Example 2 Integrating the left side:
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Example 2 (cont.) Integrating the right side: Thus:
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Linear Differential Equations The standard form of a 1 st order linear differential equation is: For example:
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Linear Differential Equations General solution: Suppose we know a function v(x) such that: Multiplying the equation by v(x) we obtain:
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Linear Differential Equations The condition on v(x) is: This leads to:
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Linear Differential Equations The last equation will be satisfied if: This is a separable equation:
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Linear Differential Equations To sum up: Where:
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Example Solution:
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Example (cont.)
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Derivative with respect to time We denote (after Newton):
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RC circuits R – Resistance (in Ohms) C – Capacitance (in Farads) I R C Current source
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RC circuits I R C The dynamical equation is:
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RC circuits Defining: We obtain: The general solution is:
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RC circuit Response to a step current:
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RC circuit Response to a step current:
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Integrate-and-Fire Neuron R – Membrane Resistance (1/conductance) C – Membrane Capacitance (in Farads) I R C inside outside ELEL Threshold mechanism
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Integrate-and-Fire Neuron If we define: The dynamical equation will be: To simplify, we define: Thus:
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Integrate-and-Fire Neuron The threshold mechanism: –For V<θ the cell obeys its passive dynamics –For V=θ the cell fires a spike and the voltage resets to 0. After voltage reset there is a refractory period, τ R.
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Integrate-and-Fire Neuron Response to a step current: IR<θ: t V
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Integrate-and-Fire Neuron Response to a step current: IR>θ: V t T τRτR τRτR τRτR
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