## SUMMER SCHOOL IN THEORETICAL PHYSICS LES HOUCHES, FRANCE THEORY OF POINT PROCESSES FOR NEURAL SYSTEMS (2003)

### BibTeX

@MISC{Brown03summerschool,

author = {Emery N. Brown},

title = {SUMMER SCHOOL IN THEORETICAL PHYSICS LES HOUCHES, FRANCE THEORY OF POINT PROCESSES FOR NEURAL SYSTEMS},

year = {2003}

}

### OpenURL

### Abstract

preparing the figures. page 2: Les Houches 2003; An Introduction to the Theory of Point Processes; Emery N. Brown 1. Neural spike trains as point processes Modeling analyses of neural systems are typically performed with Hodgkin and Huxley, integrate-and-fire and neural network models. In general, these models treat the processes of action potential production as deterministic. Much insight in the behavior of neural systems has

### Citations

547 |
The Statistical Analysis of Failure Time Data
- Kalbfleisch, Prentice
- 2002
(Show Context)
Citation Context ... λ ( t| Ht) depends on the history of the spike train and therefore, it is also termed the stochastic intensity. In survival analysis the conditional intensity function is called the hazard function. =-=[9,10]-=- This is because the hazard function can be used to define the probability of an event in the interval [ tt+∆ , ) given that there has not been an event up to t . For example it might represent the pr... |

446 |
Stochastic Simulation
- Ripley
- 1987
(Show Context)
Citation Context ...no spike if u k = 0. While in many instances there will be faster, more computationally efficient algorithms for simulating a point process, such as model based methods for specific renewal processes =-=[26]-=- and thinning algorithms, the algorithm in Eq. 7.6 is simple to implement. For an example of where this algorithm is crucial for point process simulation we consider the inhomogeneous inverse gamma mo... |

363 |
The Theory of Stochastic Processes
- Cox, Miller
- 1965
(Show Context)
Citation Context ...aluated explicitly in specific cases. 5.2.2. Asymptotic distribution of Nt () for large t . Because Eq. 5.19 is a challenge to evaluate, asymptotic approximations are frequently used. It can be shown =-=[14,15]-=- that for large t the distribution of Nt () may be approximated as a Gaussian random variable with mean and variance defined by where E[ Nt ( )] tµ − 1 2 3 = σ µ − Page 19 Var[ N( t)] = t (5.20) 2 i i... |

150 |
An Introduction to the Theory of
- Daley, Vere-Jones
- 1988
(Show Context)
Citation Context ...s is a stochastic process composed of a sequence of binary events that occur in continuous time. The theory of point processes is a highly developed subdiscipline in the field of stochastic processes =-=[1]-=-. There has recently been extensive theoretical study of point processes as well as application of this theory in biostatistics, geophysics and stochastic control. In many cases, these results have be... |

47 |
Stochastic Modeling of Scientific Data
- Guttorp
- 1995
(Show Context)
Citation Context ...0 < u1< u2, …, uj ≤ t∩ N( t) = j} , where Nt () is the number of spikes in (0, t ] and j ≤ J . The sample path is a right continuous function that jumps 1 at the spike times and is constant otherwise =-=[1,5,6,7,8]-=-. The function N 0:t tracks the location and number of spikes in (0, t ] and hence, contains all the information in the sequence of spike times (Fig. 3A). The counting process Ntgives () the total num... |

33 |
Theoretical Probability for Applications
- Port
- 1994
(Show Context)
Citation Context ... u , u , …, u ) n+ 1 T 1 2 n n+ 1 1 2 n ⎧ T ⎫ = exp ⎨− λ( u| Hu) du⎬ n ⎩ ∫un ⎭ = exp{ −τT}, where the last equality follows from the definition of τ T . By the multivariate change-of-variable formula =-=[19]-=- 1 2 n 1 2 n n (6.7) p( τ , τ , …, τ ) = | J | p( u , u , …, u ∩ N( u ) = n) , (6.8) where J is the Jacobian of the transformation between uj, j = 1, … , n and τ k , k = 1, … , n. Because τ k is a fun... |

25 | Introduction to Probability Models, 6th ed - Ross - 1997 |

20 |
1982]: Statistical analysis of counting processes
- Jacobsen
(Show Context)
Citation Context ...0 < u1< u2, …, uj ≤ t∩ N( t) = j} , where Nt () is the number of spikes in (0, t ] and j ≤ J . The sample path is a right continuous function that jumps 1 at the spike times and is constant otherwise =-=[1,5,6,7,8]-=-. The function N 0:t tracks the location and number of spikes in (0, t ] and hence, contains all the information in the sequence of spike times (Fig. 3A). The counting process Ntgives () the total num... |

16 |
Démonstration simplifiée d’un théorème de Knight
- Meyer
(Show Context)
Citation Context ...e with stationary Gaussian processes, it is possible to have two different stationary point process models with the same spectral density. 6. The time-rescaling theorem This result, originally due to =-=[16]-=- and [17], states that every point process with a conditional intensity function maps into a Poisson process with unit rate. In addition to being an interesting theoretical result, it has important im... |

11 |
Introduction to Theoretical Neurobiology: Nonlinear and Stochastic Theories
- Tuckwell
- 1988
(Show Context)
Citation Context ... In Section 3, we develop the characterization of a point process in terms of its conditional intensity function and we relate it to the interspike interval distribution. This section follows closely =-=[2]-=-. 2.1. Non-leaky integrator with excitatory Poisson inputs Consider a neuron whose membrane voltage time course is defined by dV () t = αE dN(), t (2.1) where Ntis () a Poisson process with constant r... |

6 |
A �rst course in real analysis (2nd ed
- Protter, Morrey
- 1991
(Show Context)
Citation Context ...ents defined as | J | = | ∏ Jkk | . By assumption 0 < λ( t| Ht) and, by Eq. 6.4 and the k= 1 definition of τ k the mapping of u into τ is one-to-one. Therefore, by the inverse differentiation theorem =-=[20]-=- the diagonal elements of J are ∂uk Jkk = = λ( uk| Hu) k ∂τ k −1 . (6.9)spage 22: Les Houches 2003; An Introduction to the Theory of Point Processes; Emery N. Brown Substituting | J | and Eq. 6.2 into... |

1 |
The Inverse Gaussian Distribution: Theory
- Chhikara, Folks
- 1989
(Show Context)
Citation Context ... a Wiener process with drift then, what is the interspike interval probability density? We define the first passage time as the condition t = inf{ u| V( u) = θ} , V(0) = V0< θ . We consider two cases =-=[2,3]-=-. First θ we start with the driftless Wiener process, that is with β = 0 . For the driftless Wiener process model the first passage time probability density is 2 V ⎧ 0 ⎪ ( V0) ⎫⎪ ⎨ 23 2 ⎬ θ − θ − pθ()... |

1 |
105 (2001) 25-37. Page 31 32: Les Houches 2003; An Introduction to the Theory of Point Processes; Emery N. Brown A Threshold Membrane Potential Threshold Membrane Potential Spike Train B Spike Train 0 0.5 Time (sec
- Meth
(Show Context)
Citation Context ...inning algorithms, the algorithm in Eq. 7.6 is simple to implement. For an example of where this algorithm is crucial for point process simulation we consider the inhomogeneous inverse gamma model in =-=[27]-=-. Its conditional intensity function is infinite immediately following a spike ifψ < 1. If in addition, ψ is time-varying, i.e., ψ = ψ() t < 1 for all t , then neither thinning nor standard algorithms... |