Importance of the Cutoff Value in the Quadratic Adaptive Integrate-and-Fire Model

The quadratic adaptive integrate-and-fire model (Izhikevich, 2003 , 2007 ) is able to reproduce various firing patterns of cortical neurons and is widely used in large-scale simulations of neural networks. This model describes the dynamics of the membrane potential by a differential equation that is quadratic in the voltage, coupled to a second equation for adaptation. Integration is stopped during the rise phase of a spike at a voltage cutoff value Vc or when it blows up. Subsequently the membrane potential is reset, and the adaptation variable is increased by a fixed amount. We show in this note that in the absence of a cutoff value, not only the voltage but also the adaptation variable diverges in finite time during spike generation in the quadratic model. The divergence of the adaptation variable makes the system very sensitive to the cutoff: changing Vc can dramatically alter the spike patterns. Furthermore, from a computational viewpoint, the divergence of the adaptation variable implies that the time steps for numerical simulation need to be small and adaptive. However, divergence of the adaptation variable does not occur for the quartic model (Touboul, 2008 ) and the adaptive exponential integrate-and-fire model (Brette & Gerstner, 2005 ). Hence, these models are robust to changes in the cutoff value.

Download Full-text

Generalized Integrate-and-Fire Models of Neuronal Activity Approximate Spike Trains of a Detailed Model to a High Degree of Accuracy

Journal of Neurophysiology ◽

10.1152/jn.00190.2004 ◽

2004 ◽

Vol 92 (2) ◽

pp. 959-976 ◽

Cited By ~ 158

Author(s):

Renaud Jolivet ◽

Timothy J. Lewis ◽

Wulfram Gerstner

Keyword(s):

Cortical Neurons ◽

Direct Reduction ◽

Spike Trains ◽

Model Parameters ◽

Detailed Model ◽

Fire Model ◽

Integrate And Fire ◽

Fire Models ◽

Wide Range ◽

Simple Models

We demonstrate that single-variable integrate-and-fire models can quantitatively capture the dynamics of a physiologically detailed model for fast-spiking cortical neurons. Through a systematic set of approximations, we reduce the conductance-based model to 2 variants of integrate-and-fire models. In the first variant (nonlinear integrate-and-fire model), parameters depend on the instantaneous membrane potential, whereas in the second variant, they depend on the time elapsed since the last spike [Spike Response Model (SRM)]. The direct reduction links features of the simple models to biophysical features of the full conductance-based model. To quantitatively test the predictive power of the SRM and of the nonlinear integrate-and-fire model, we compare spike trains in the simple models to those in the full conductance-based model when the models are subjected to identical randomly fluctuating input. For random current input, the simple models reproduce 70–80 percent of the spikes in the full model (with temporal precision of ±2 ms) over a wide range of firing frequencies. For random conductance injection, up to 73 percent of spikes are coincident. We also present a technique for numerically optimizing parameters in the SRM and the nonlinear integrate-and-fire model based on spike trains in the full conductance-based model. This technique can be used to tune simple models to reproduce spike trains of real neurons.

Download Full-text

The Ornstein-Uhlenbeck Process Does Not Reproduce Spiking Statistics of Neurons in Prefrontal Cortex

Neural Computation ◽

10.1162/089976699300016511 ◽

1999 ◽

Vol 11 (4) ◽

pp. 935-951 ◽

Cited By ~ 79

Author(s):

Shigeru Shinomoto ◽

Yutaka Sakai ◽

Shintaro Funahashi

Keyword(s):

Prefrontal Cortex ◽

Order Statistics ◽

Cortical Neurons ◽

Biological Data ◽

Higher Order Statistics ◽

Uhlenbeck Process ◽

Fire Model ◽

Integrate And Fire ◽

Spike Sequences ◽

Ornstein Uhlenbeck Process

Cortical neurons of behaving animals generate irregular spike sequences. Recently, there has been a heated discussion about the origin of this irregularity. Softky and Koch (1993) pointed out the inability of standard single-neuron models to reproduce the irregularity of the observed spike sequences when the model parameters are chosen within a certain range that they consider to be plausible. Shadlen and Newsome (1994), on the other hand, demonstrated that a standard leaky integrate-and-fire model can reproduce the irregularity if the inhibition is balanced with the excitation. Motivated by this discussion, we attempted to determine whether the Ornstein-Uhlenbeck process, which is naturally derived from the leaky integration assumption, can in fact reproduce higher-order statistics of biological data. For this purpose, we consider actual neuronal spike sequences recorded from the monkey prefrontal cortex to calculate the higher-order statistics of the interspike intervals. Consistency of the data with the model is examined on the basis of the coefficient of variation and the skewness coefficient, which are, respectively, a measure of the spiking irregularity and a measure of the asymmetry of the interval distribution. It is found that the biological data are not consistent with the model if the model time constant assumes a value within a certain range believed to cover all reasonable values. This fact suggests that the leaky integrate-and-fire model with the assumption of uncorrelated inputs is not adequate to account for the spiking in at least some cortical neurons.

Download Full-text

Robust Modulation of Integrate-and-Fire Models

Neural Computation ◽

10.1162/neco_a_01065 ◽

2018 ◽

Vol 30 (4) ◽

pp. 987-1011 ◽

Cited By ~ 5

Author(s):

Tomas Van Pottelbergh ◽

Guillaume Drion ◽

Rodolphe Sepulchre

Keyword(s):

Large Scale ◽

Neuronal Excitability ◽

Population Studies ◽

Fire Model ◽

Neuronal Populations ◽

Integrate And Fire ◽

Large Scale Network ◽

Channel Expression ◽

Scale Network ◽

Network Simulations

By controlling the state of neuronal populations, neuromodulators ultimately affect behavior. A key neuromodulation mechanism is the alteration of neuronal excitability via the modulation of ion channel expression. This type of neuromodulation is normally studied with conductance-based models, but those models are computationally challenging for large-scale network simulations needed in population studies. This article studies the modulation properties of the multiquadratic integrate-and-fire model, a generalization of the classical quadratic integrate-and-fire model. The model is shown to combine the computational economy of integrate-and-fire modeling and the physiological interpretability of conductance-based modeling. It is therefore a good candidate for affordable computational studies of neuromodulation in large networks.

Download Full-text

An integrate-and-fire model for synchronized bursting in a network of cultured cortical neurons

Journal of Computational Neuroscience ◽

10.1007/s10827-006-7815-5 ◽

2006 ◽

Vol 21 (3) ◽

pp. 227-241 ◽

Cited By ~ 16

Author(s):

D. A. French ◽

E. I. Gruenstein

Keyword(s):

Cortical Neurons ◽

Fire Model ◽

Integrate And Fire ◽

Cultured Cortical Neurons

Download Full-text

On the Simulation of Nonlinear Bidimensional Spiking Neuron Models

Neural Computation ◽

10.1162/neco_a_00141 ◽

2011 ◽

Vol 23 (7) ◽

pp. 1704-1742 ◽

Cited By ~ 4

Author(s):

Jonathan Touboul

Keyword(s):

Differential Equation ◽

Membrane Potential ◽

Cortical Neurons ◽

Fixed Time ◽

Computational Time ◽

Integration Step ◽

Large Network ◽

Time Step ◽

Integration Schemes ◽

Adaptation Variable

Bidimensional spiking models are garnering a lot of attention for their simplicity and their ability to reproduce various spiking patterns of cortical neurons and are used particularly for large network simulations. These models describe the dynamics of the membrane potential by a nonlinear differential equation that blows up in finite time, coupled to a second equation for adaptation. Spikes are emitted when the membrane potential blows up or reaches a cutoff θ. The precise simulation of the spike times and of the adaptation variable is critical, for it governs the spike pattern produced and is hard to compute accurately because of the exploding nature of the system at the spike times. We thoroughly study the precision of fixed time-step integration schemes for this type of model and demonstrate that these methods produce systematic errors that are unbounded, as the cutoff value is increased, in the evaluation of the two crucial quantities: the spike time and the value of the adaptation variable at this time. Precise evaluation of these quantities therefore involves very small time steps and long simulation times. In order to achieve a fixed absolute precision in a reasonable computational time, we propose here a new algorithm to simulate these systems based on a variable integration step method that either integrates the original ordinary differential equation or the equation of the orbits in the phase plane, and compare this algorithm with fixed time-step Euler scheme and other more accurate simulation algorithms.

Download Full-text

How Sample Paths of Leaky Integrate-and-Fire Models Are Influenced by the Presence of a Firing Threshold

Neural Computation ◽

10.1162/neco_a_00143 ◽

2011 ◽

Vol 23 (7) ◽

pp. 1743-1767 ◽

Cited By ~ 7

Author(s):

Maria Teresa Giraudo ◽

Priscilla E. Greenwood ◽

Laura Sacerdote

Keyword(s):

Membrane Potential ◽

Diffusion Process ◽

Fixed Time ◽

Firing Time ◽

Fire Model ◽

Integrate And Fire ◽

Sample Paths ◽

The Mean ◽

First Time

Neural membrane potential data are necessarily conditional on observation being prior to a firing time. In a stochastic leaky integrate-and-fire model, this corresponds to conditioning the process on not crossing a boundary. In the literature, simulation and estimation have almost always been done using unconditioned processes. In this letter, we determine the stochastic differential equations of a diffusion process conditioned to stay below a level S up to a fixed time t1 and of a diffusion process conditioned to cross the boundary for the first time at t1. This allows simulation of sample paths and identification of the corresponding mean process. Differences between the mean of free and conditioned processes are illustrated, as well as the role of noise in increasing these differences.

Download Full-text