Linear Time-invarian sysems, Convoluion, and Cross-correlaion (1) Linear Time-invarian (LTI) sysem A sysem akes in an inpu funcion and reurns an oupu funcion. x() T y() Inpu Sysem Oupu y() = T[x()] An LTI sysem is a special ype of sysem. As he name suggess, i mus be boh linear and ime-invarian, as defined below. LINEAR Scaling: T[ax ()]=at[x ()] Superposiion: T[x 1 ()+x 2 ()]=T[x 1 ()]+T[x 2 ()] TIME INVARIANT If y()=t[x()], hen y(-s) =T[x(-s)] Many sysems in neuroscience can be approximaed as LTI: Inpu LTI sysem Oupu Pre-synapic acion poenials Visual simulus Simulus conras Injeced Curren Synapse Reinal ganglion cell Passive neural membrane Reinal image Firing rae Membrane poenial The reason LTI sysems are incredibly useful is because of a key fac: if you know he response of he sysem o an impulse, han you can calculae he response of he sysem o ANY inpu. This gives you an enormous amoun of predicive power. Bu wha do I mean by an impulse? As ploed in he figure below, he impulse funcion is jus a funcion ha is zero a all imes excep for a =0 (lef panel below). You can hink of i as he force on an objec creaed by a hammer hiing i he force is 0 excep for a he momen of impac. The impulse response is he response of he sysem o he impulse. You can hink of his as he response of your leg o your knee geing hi wih a hammer a he docor s office. The righ panel below is an example of wha he impulse response of a sysem migh look like. Eye Pos-synapic conducance
The Impulse funcion: 1, δ ( ) = 0, δ() = 0 0 The Impulse response: T h() (2) Convoluion How do we predic he response of our sysem o an arbirary inpu once we know he impulse response? The secre is convoluion. I allows us o predic he response of our knee o some arbirary inpu funcion (once we know he response o he hammer), or he response of a building o an earhquake (once we know how i responds o he impac of an impulse). Convoluion is a mahemaical operaion which akes wo funcions and produces a hird funcion ha represens he amoun of overlap beween one of he funcions and a reversed and ranslaed version of he oher funcion. Here is he mahemaical definiion of convolving wo funcions, x() and h(), o creae an oupu y(): ( ) y = x( τ ) h( τ ) τ = This is ofen wrien in shorhand as y=x*h, where he * represens he convoluion operaion. You can hink of in he equaion above as a variable ha represens ime. Bu wha is τ? I s called a dummy variable. As you can see, i doesn exis on he lef hand side of he equaion i s jus used o index over ime for he inegraion on he righ hand side. Hopefully, he example below of convolving sample funcions x() and h() will make his clear 1. Here are our funcions x() and h(). 1 Example borrowed from hp://www.ee.washingon.edu/class/235dl/ee235/projec/lesson6/lesson6_1.hml
The firs sep is o reverse one of he funcions. We choose o reverse h(), which we now plo on he dummy ime axis τ. We plo x(τ) on he same axis, and begin he process of shifing h(-τ) by, and comparing i o x(τ). Since hese are coninuous (no discree) funcions, we ake an inegral (no he sum) when calculaing he convoluion. In he figure below, h is shifed by =-2. For his value of shif, here is no overlap beween x(τ) and h(-τ), so y()=0 (which is why i doesn show up in he plo below). As we coninue o shif h(-τ) by changing, here ges o be overlap beween he wo funcions, as ploed below (he convoluion, y(), is depiced in pink). The overlap is quanified by muliplying he wo inpu funcions on a poin by poin basis, and hen inegraing he resuling funcion.
Once reaches 5, here is no longer overlap beween he wo funcions, and y()=0 once again. Convoluion is an incredibly useful operaion because i can be used o predic he oupu of an LTI sysem o any inpu. Jus hink of x() as he arbirary inpu funcion (e.g. a visual simulus) and h() as he response of he sysem o an impulse (e.g. he firing rae of he reinal ganglion cell o a flash of ligh). Then he convoluion of x() and h() is he prediced oupu of he sysem (e.g. he firing rae in response o he arbirary visual simulus). Because of his grea prediciive power, LTI sysems are used all he ime in neuroscience. A useful hing o know abou convoluion is he Convoluion Theorem, which saes ha convolving wo funcions in he ime domain is he same as muliplying hem in he frequency domain: If y()=x()*h(), (remember, * means convoluion) hen Y(f)=X(f)H(f) (where Y is he fourier ransform of y, X is he fourier ransform of x, ec) Since muliplying is a simpler operaion ha convolving, i is ofen easier o work in he frequency domain. (3) Cross-correlaion Cross-correlaion is a mahemaical operaion ha is used o quanify he similariy of wo funcions for differen ime delays. For example, i can be used o compare firing rae beween wo neurons o see a wha relaive ime delay he firing is mos similar. Despie having a differen purpose han convoluion, he mahemaical operaion of cross-correlaion is very similar o ha of convoluion. Mahemaically, he main difference is ha when wo funcions are cross-correlaed, neiher funcion is imereversed before being shifed and compared. Cross-correlaion f _ g = f ( τ ) g( + τ ) dτ Convoluion f g = f ( τ ) g( τ ) dτ
As a simple example, le s consider he cross-correlaion of wo neurons wih he following spiking paern (1 represens a spike during he imebin; 0 represens no spike). ime Neuron 1 spike rain: 0 0 1 0 1 0 0 0 1 1 0 0 0 1 1 Neuron 2 spike rain: 0 0 0 1 0 1 0 0 0 1 1 0 0 0 1 To calculae he cross-correlaion, we need o compare he spike rains for differen relaive ime delay (i.e. differen relaive shifs of he wo spike rains). Firs, le s consider a ime delay of 0. When we muliply he wo spike rains on a poin by poin basis, we ge he following: 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 The nex sep of calculaing he cross correlaion is o sum across all hese values. This gives us a cross-correlaion of 2 (for a ime delay of 0). We repea he same calculaion for differen delays. As you can see from he plo below, he cross correlaion peaks for ime delay of -1. Crosscorr -2-1 0 1 2 3 Time delay The inerpreaion of his cross-correlaion is ha he wo spike rains are in he greaes agreemen when spike rain 2 is shifed backwards in ime by one ime poin. In oher words, neuron 2 ends o spike afer neuron 1. A biological scenario ha could lead o his cross-correlaion profile is ha neuron 2 is possynapic o neuron 1.