Introduction to Mathematical Methods in Neurobiology: Dynamical Systems

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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 equationsDescribe the evolution of
  systems in continuous time.
• Difference equations / Iterated mapsDescribe
  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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1st Order Differential Equations

• A 1st 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 1st 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
Current source

• R Resistance (in Ohms)
• C Capacitance (in Farads)

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RC circuits

• 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)

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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 Vlt? the cell obeys its passive dynamics
• For V? the cell fires a spike and the voltage
  resets to 0.
• After voltage reset there is a refractory period,
  tR.

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Integrate-and-Fire Neuron

• Response to a step current
• IRlt?

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Integrate-and-Fire Neuron

• Response to a step current
• IRgt?