Lecture 2: Everything you need to know to know about point processes Outline: basic ideas homogeneous (stationary) Poisson processes Poisson distribution inter-event interval distribution coefficient of variance (CV) correlation function stationary renewal process relation between IEI distribution and correlation function Fano factor F relation between F and CV nonstationary (inhomogeneous) Poisson process time rescaling

Point Processes Point process: discrete set of points (events) on the real numbers (or some interval on the reals)

Point Processes Point process: discrete set of points (events) on the real numbers (or some interval on the reals) Usually we are thinking of times of otherwise identical events. (but sometimes space or space-time)

Point Processes Point process: discrete set of points (events) on the real numbers (or some interval on the reals) Usually we are thinking of times of otherwise identical events. (but sometimes space or space-time) Examples: radioactive decay, arrival times, earthquakes, neuronal spike trains, …

Point Processes Point process: discrete set of points (events) on the real numbers (or some interval on the reals) Usually we are thinking of times of otherwise identical events. (but sometimes space or space-time) Examples: radioactive decay, arrival times, earthquakes, neuronal spike trains, … Stochastic: characterized by the probability (density) of every set {t 1, t 2, … t N }

Neuronal spike trains Action potential:

Neuronal spike trains spike trains evoked by many presentations of the same stimulus: Action potential:

Neuronal spike trains spike trains evoked by many presentations of the same stimulus: Action potential: (apparently) stochastic

Homogeneous Poisson process

Homogeneous Poisson process: r = rate = prob of event per unit time, i.e., rΔt = prob of event in interval [t, t + Δt) (Δt  0)

Homogeneous Poisson process Homogeneous Poisson process: r = rate = prob of event per unit time, i.e., rΔt = prob of event in interval [t, t + Δt) (Δt  0) Survivor function: probability of no event in [0,t): S(t)

Homogeneous Poisson process Homogeneous Poisson process: r = rate = prob of event per unit time, i.e., rΔt = prob of event in interval [t, t + Δt) (Δt  0) Survivor function: probability of no event in [0,t): S(t)

Homogeneous Poisson process Homogeneous Poisson process: r = rate = prob of event per unit time, i.e., rΔt = prob of event in interval [t, t + Δt) (Δt  0) Survivor function: probability of no event in [0,t): S(t) Probability /unit time of first event in [t, t +  t)) :

Homogeneous Poisson process Homogeneous Poisson process: r = rate = prob of event per unit time, i.e., rΔt = prob of event in interval [t, t + Δt) (Δt  0) Survivor function: probability of no event in [0,t): S(t) Probability /unit time of first event in [t, t +  t)) :

Homogeneous Poisson process Homogeneous Poisson process: r = rate = prob of event per unit time, i.e., rΔt = prob of event in interval [t, t + Δt) (Δt  0) Survivor function: probability of no event in [0,t): S(t) Probability /unit time of first event in [t, t +  t)) : (inter-event interval distribution)

Homogeneous Poisson process (2) Probability of exactly 1 event in [0,T):

Homogeneous Poisson process (2) Probability of exactly 1 event in [0,T): Probability of exactly 2 events in [0,T):

Homogeneous Poisson process (2) Probability of exactly 1 event in [0,T): Probability of exactly 2 events in [0,T): … Probability of exactly n events in [0,T):

Homogeneous Poisson process (2) Probability of exactly 1 event in [0,T): Probability of exactly 2 events in [0,T): … Probability of exactly n events in [0,T): Poisson distribution

Probability of n events in interval of duration T :

Poisson distribution Probability of n events in interval of duration T : mean count: = rT

Poisson distribution Probability of n events in interval of duration T : mean count: = rT variance: ) 2 > = rT, i.e. ± 1/2

Poisson distribution Probability of n events in interval of duration T : mean count: = rT variance: ) 2 > = rT, i.e. ± 1/2 large rT :  Gaussian

Poisson distribution Probability of n events in interval of duration T : mean count: = rT variance: ) 2 > = rT, i.e. ± 1/2 large rT :  Gaussian

Characteristic function Poisson distribution with mean a :

Characteristic function Poisson distribution with mean a : Characteristic function

Poisson distribution with mean a : Characteristic function

Poisson distribution with mean a : Characteristic function Cumulant generating function

Characteristic function Poisson distribution with mean a : Characteristic function Cumulant generating function

Characteristic function Poisson distribution with mean a : Characteristic function Cumulant generating function

Characteristic function Poisson distribution with mean a : Characteristic function Cumulant generating function All cumulants = a

Homogeneous Poisson process (3): inter-event interval distribution Exponential distribution: (like radioactive Decay)

Homogeneous Poisson process (3): inter-event interval distribution Exponential distribution: (like radioactive Decay) mean IEI:

Homogeneous Poisson process (3): inter-event interval distribution Exponential distribution: (like radioactive Decay) mean IEI: variance:

Homogeneous Poisson process (3): inter-event interval distribution Exponential distribution: (like radioactive Decay) mean IEI: variance: Coefficient of variation:

Homogeneous Poisson process (4): correlation function notation:

Homogeneous Poisson process (4): correlation function notation:

Homogeneous Poisson process (4): correlation function notation: mean:

Homogeneous Poisson process (4): correlation function notation: mean: correlation function:

Proof: (1):

Proof: (1): S(t) and S(t’) are independent, so

Proof: Use finite-width, finite-height delta functions: (1): S(t) and S(t’) are independent, so (2):

Proof: Use finite-width, finite-height delta functions: (1): S(t) and S(t’) are independent, so (2):

Proof: Use finite-width, finite-height delta functions: (1): S(t) and S(t’) are independent, so (2):

Proof: Use finite-width, finite-height delta functions: (1): S(t) and S(t’) are independent, so (2): so

Proof: Use finite-width, finite-height delta functions: (1): S(t) and S(t’) are independent, so (2): so

Proof: Use finite-width, finite-height delta functions: (1): S(t) and S(t’) are independent, so (2): so

Proof: Use finite-width, finite-height delta functions: (1): S(t) and S(t’) are independent, so (2): so

General stationary point process: For any point process,

General stationary point process: For any point process, with A(t) continuous,

General stationary point process: For any point process, with A(t) continuous, (because S(t) is composed of delta-functions)

Stationary renewal process Defined by ISI distribution P(t)

Stationary renewal process Defined by ISI distribution P(t) Relation between P(t) and C(t) :

Stationary renewal process Defined by ISI distribution P(t) Relation between P(t) and C(t) : define

Stationary renewal process Defined by ISI distribution P(t) Relation between P(t) and C(t) : define

Stationary renewal process Defined by ISI distribution P(t) Relation between P(t) and C(t) : define

Stationary renewal process Defined by ISI distribution P(t) Relation between P(t) and C(t) : define Laplace transform:

Stationary renewal process Defined by ISI distribution P(t) Relation between P(t) and C(t) : define Laplace transform: Solve:

Fano factor spike count variance / mean spike count

Fano factor spike count variance / mean spike count for stationary Poisson process

Fano factor spike count variance / mean spike count for stationary Poisson process

Fano factor spike count variance / mean spike count for stationary Poisson process  t 2 t 1 

Fano factor spike count variance / mean spike count for stationary Poisson process  τ t 

Fano factor spike count variance / mean spike count for stationary Poisson process  τ t 

F=CV 2 for a stationary renewal process

F depends on integral of C(t), CV depends on moments of P(t).

F=CV 2 for a stationary renewal process F depends on integral of C(t), CV depends on moments of P(t). Use relation between C(λ) and P(λ) to relate F and CV.

F=CV 2 for a stationary renewal process F depends on integral of C(t), CV depends on moments of P(t). Use relation between C(λ) and P(λ) to relate F and CV. The algebra:

F=CV 2 for a stationary renewal process F depends on integral of C(t), CV depends on moments of P(t). Use relation between C(λ) and P(λ) to relate F and CV. The algebra:

F=CV 2 for a stationary renewal process F depends on integral of C(t), CV depends on moments of P(t). Use relation between C(λ) and P(λ) to relate F and CV. The algebra:

F=CV 2 for a stationary renewal process F depends on integral of C(t), CV depends on moments of P(t). Use relation between C(λ) and P(λ) to relate F and CV. The algebra:

F=CV 2 for a stationary renewal process F depends on integral of C(t), CV depends on moments of P(t). Use relation between C(λ) and P(λ) to relate F and CV. The algebra: Now use

F=CV 2 for a stationary renewal process F depends on integral of C(t), CV depends on moments of P(t). Use relation between C(λ) and P(λ) to relate F and CV. The algebra: Now use and

F=CV 2 (continued)

Nonstationary Poisson processes Nonstationary Poisson process: time-dependent rate r(t)

Nonstationary Poisson processes Nonstationary Poisson process: time-dependent rate r(t) Time rescaling: instead of t, use

Nonstationary Poisson processes Nonstationary Poisson process: time-dependent rate r(t) Time rescaling: instead of t, use Then the event rate per unit s is 1, i.e. the process is stationary when viewed as a function of s.

Nonstationary Poisson processes Nonstationary Poisson process: time-dependent rate r(t) Time rescaling: instead of t, use Then the event rate per unit s is 1, i.e. the process is stationary when viewed as a function of s. In particular, still have Poisson count distribution in any interval

Nonstationary Poisson processes Nonstationary Poisson process: time-dependent rate r(t) Time rescaling: instead of t, use Then the event rate per unit s is 1, i.e. the process is stationary when viewed as a function of s. In particular, still have Poisson count distribution in any interval F =1

Nonstationary Poisson processes Nonstationary Poisson process: time-dependent rate r(t) Time rescaling: instead of t, use Then the event rate per unit s is 1, i.e. the process is stationary when viewed as a function of s. In particular, still have Poisson count distribution in any interval F =1 Nonstationary renewal process: time-dependent inter-event distribution

Nonstationary Poisson processes Nonstationary Poisson process: time-dependent rate r(t) Time rescaling: instead of t, use Then the event rate per unit s is 1, i.e. the process is stationary when viewed as a function of s. In particular, still have Poisson count distribution in any interval F =1 Nonstationary renewal process: time-dependent inter-event distribution = inter-event probability starting at t 0

homework Prove that the ISI distribution is exponential for a stationary Poisson process. Prove that the CV is 1 for a stationary Poisson process. Show that the Poisson distribution approaches a Gaussian one for large mean spike count. Prove that F = CV 2 for a stationary renewal process. Show why the spike count distribution for an inhomogeneous Poisson process is the same as that for a homogeneous Poisson process with the same mean spike count.