Lecture IV: LTI models of physical systems

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Buddy Parks
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1 BME 171: Signals and Systems Duke University September 5, 2008

2 This lecture Plan for the lecture: 1 Interconnections of linear systems 2 Differential equation models of LTI systems 3 eview of linear circuit theory resistors, inductors, capacitors Kirchhoff s laws 4 Examples of LC circuits 5 Leaky integrate-and-fire (LIF) neuron

3 Interconnections of linear systems Linearity is preserved when systems are interconnected. { } { } cascade: S = S 2 {S } 1 w(t) S 1 S 2 S { sum: S } { } { } = S 1 S 2 S 1 S 2 S { } { } feedback S = S 1 { } S 2 w(t) - S 1 S 2 S

4 Cascade w(t) S 1 S 2 S Let w(t) be the output of S 1. Then we first use linearity of S 1 : } { } { } w(t) = S 1 {a 1 x 1 (t) a 2 x 2 (t) = a 1 S 1 x 1 (t) a 2 S 1 x 2 (t) Now use linearity of S 2 : { } { } { S a 1 x 1 (t) a 2 x 2 (t) = S 2 w(t) = S 2 {a 1 S 1 x1 (t) } { a 2 S 1 x2 (t) }} { { = a 1 S 2 S 1 x1 (t) }} { { a 2 S 2 S 1 x2 (t) }} { } { } = a 1 S x 1 (t) a 2 S x 2 (t) This proves that S is linear.

5 Sum S 1 S 2 { } S a 1 x 1 (t) a 2 x 2 (t) } { } = S 1 {a 1 x 1 (t) a 2 x 2 (t) S 2 a 1 x 1 (t) a 2 x 2 (t) { = a 1 S 1 x1 (t) } { a 2 S 1 x2 (t) } { a 1 S 2 x1 (t) } { a 2 S 2 x2 (t) } }{{}}{{} use linearityof S 1 use linearityof S 2 ( { = a 1 S1 x1 (t) } { S 2 x1 (t) }) ( { a 2 S1 x2 (t) } { S 2 x2 (t) }) }{{}}{{} { } { } =S x 1(t) = a 1 S { x 1 (t) } a 2 S { x 2 (t) } This proves that S is linear. S =S x 2(t)

6 Feedback w(t) - S 1 { } { } Let w(t) = S 2. Now, = S1 w(t), so { } w(t) = S 2 {S } { } 1 w(t) = S w(t), S 2 S where S is the cascade of S 1 and S 2, which is linear if both S 1 and S 2 are. The system S 3 with input and output w(t), defined by { } w(t) = S w(t), is linear. Thus, } { } = S 1 {w(t) = S 1 {S } 3 is a cascade of S 3 and S 1, and so is linear.

7 LTI systems via differential equations A lot of continuous-time LTI systems are described by linear differential equations with constant coefficients: M m=0 a m d m m = N n=0 b n d n n where the coefficients {a m } M m=1 and {b n} N N=1 are independent of t. Examples: linear electric circuits (LC) mechanical systems (mass-spring-damper) We will focus on electrical circuits.

8 eview: linear circuit elements esistor: i(t) v(t) Inductor: i(t) L v(t) Capacitor: i(t) C v(t) v(t) = i(t) i(t) = v(t) v(t) = L di(t) i(t) = 1 L t v(τ)dτ i(t) = C dv(t) v(t) = 1 C t i(τ)dτ

9 eview: Kirchhoff s laws Kirchhoff s voltage law (KVL): Kirchhoff s current law (KCL): v 2 _ i 3 i 4 v 1 v 3 i 2 i 1 The sum of voltages in a loop is equal to zero: v 1 v 2 v 3 = 0 The sum of currents entering a node is equal to zero: i 1 i 2 i 3 i 4 = 0

10 Example: Series LC circuit L _ i(t) C _ Input: voltage source Output: voltage across the capacitor Apply KVL: i(t) L di(t) Substitute i(t) = C d : C d earrange to get LC d2 2 = 0 LC d2 2 = 0 C d =

11 Example: Parallel C circuit i (t) i C (t) C _ Input: current source Output: voltage across the capacitor Apply KCL: Substitute i (t) = earrange to get = i (t) i C (t) and i C(t) = C d : = C d C d 1 =

12 Example: biological neurons Biological neurons are highly nonlinear systems that convert incoming electrical signals (encoding external stimuli) into spike trains: neuron 0 t 0 t Inputs to the neuron are electrical signals traveling along the dendrites to the body (or soma) of the neuron. The neuron accumulates a potential (voltage) across its cell membrane and then fires, i.e., emits an electric pulse that travels down the axon.

13 Leaky integrate-and-fire (LIF) neuron The leaky integrate-and-fire (LIF) neuron is a simple model that describes the salient features of biological neurons. The LIF neuron has two distinct operating regimes: subthreshold when the membrane potential of the neuron is below a certain threshold value V th, the neuron acts like a parallel C circuit. The capacitance is due to charge buildup on both sides of the bilipid layer that forms the cell membrane; the resistance is due to the presence of protein channels in the membrane that can carry K, Na and Cl ions in and out of the cell (leakage current) superthreshold when the membrane potential crosses V th, the neuron fires (emits a unit impulse), and then short-circuits for τ ref seconds (the time known as the refractory period). After the refractory period elapses, the neuron returns to the subthreshold regime.

14 Circuit model of the subthreshold regime Let us look at the subthreshold regime of the LIF neuron with a unit step input = u(t) i (t) i C (t) C _ C d = 1 = u(t) [ 1 e t/c] u(t) The overall output of the LIF neuron due to the unit step input looks like this: V th 0 τ ref t

Kirchhoff's Laws and Circuit Analysis (EC 2)

Kirchhoff's Laws and Circuit Analysis (EC ) Circuit analysis: solving for I and V at each element Linear circuits: involve resistors, capacitors, inductors Initial analysis uses only resistors Power sources,