An event-based integration scheme for an integrate-and-fire neuron model with exponentially decaying excitatory synaptic currents and double exponential inhibitory synaptic currents has recently been introduced by Carnevale and Hines. This integration scheme imposes non-physiological constraints on the time constants of the synaptic currents it attempts to model which hamper the general applicability. This paper addresses this problem in two ways. First, we provide physical arguments to show why these constraints on the time constants can be relaxed. Second, we give a formal proof showing which constraints can be abolished. This proof rests on a generalization of the Carnevale-Hines lemma, which is a new tool for comparing double exponentials as they naturally occur in many cascaded decay systems including receptor-neurotransmitter dissociation followed by channel closing. We show that this lemma can be generalized and subsequently used for lifting most of the original constraints on the time constants. Thus we show that the Carnevale-Hines integration scheme for the integrate-and-fire model can be employed for simulating a much wider range of neuron and synapse type combinations than is apparent from the original treatment.
One of the most salient features of neurons is their ability to summate synaptic inputs arriving from other neurons and to respond with the generation of an action potential or spike when the membrane potential reaches a certain threshold value. After its generation, a spike will generally travel down the neurons axon to serve as an input to other cells, including muscles fibers and neurons. In its most basic form, spike generation is captured by the so-called integrate-and-fire model. This model was first conceived a hundred years ago by Lapicque [1]. Lapicque modeled the subthreshold behavior of the membrane potential as a capacitance in parallel with a resistor based on the electrical properties of the cell membrane. At that time, the spike generating mechanism was not known, and it was therefore only possible to give a phenomenological description of the process. On the basis of electrophysiological experiments, Lapicque assumed that when the membrane potential reached a threshold value, the cell would generate (fire) a spike and subsequently the membrane potential would be reset to resting level [1,2]. Integrate-and-fire models are still widely used today, both in simulations and for the analytical study of neural network dynamics.
The integration scheme we analyze here was introduced by Carnevale and Hines [3,4] for the widely used NEURON simulation environment [5] and is event-based. In event-based models the synaptic coupling between neurons is mediated by events. Events are triggered by threshold crossings of the integrate-and-fire neurons and subsequently communicated to the postsynaptic cells. Postsynaptically these events initiate a change in the synapse, which in the most common case lead to excitation or inhibition of the postsynaptic cell. Support for event-based integration methods is available in several other scientific neural simulators ( e.g. NEST, XPP and Mvaspike) [6] which makes these simulators possible candidates for the implementation of the scheme discussed here. Besides the Carnevale-Hines scheme many other integration schemes are in use to simulate integrate-and-fire models. These schemes range from purely numerical integration schemes, such as Euler and Runge-Kutta, to numerically exact calculations based on root-finding algorithms for determining when the membrane potential crosses the spiking threshold [7,8,9,10,11,12]. The Carnevale-Hines scheme takes a middle ground between the two above mentioned extremes; it uses explicit knowledge of the exact solution to determine whether threshold crossing will occur, but avoids the expensive explicit calculation of the threshold passage time. Instead, the model employs the computationally cheaper Newton iteration to obtain a spike time estimate. For situations where presynaptic cells become active between two firing times and the order of firing of the cells is unknown (for example, due to mild external noise), we expect this scheme to be computational efficient. An analysis of computational efficiency, however, is outside the scope of this paper.
The Carnevale-Hines scheme is developed to solve an integrate-and-fire model which includes excitatory synapse as exponentially decaying currents and inhibitory synapse as currents following a double exponential function. With these currents, it is in principle possible to describe the main class of excitatory synapses (characterized by AMPAreceptors) and the main class of inhibitory synapses (characterized by GABA receptors). As mentioned before the Carnevale-Hines scheme uses Newton iteration to estimate the threshold crossing times,or phrased differently to find the events. In their proof of the correctness of the Newton iteration estimate Carnevale and Hines used the following constraints on the time constants of the synapses and the membrane time constants: τ decay,excitatory < τ rise,inhibitory < τ decay,inhibitory < τ membrane [3,4]. These constraints imposed by the integration method lead to the loss of many physiological relevant realizations of the conceptual model. To elucidate the biological relevance of this point we will discuss the physiological parameter range in the next paragraph.
In cortical areas of mammals, the excitatory AMPA currents have a fast rise time between 0.1 and 0.8 ms, followed by a fast decay of 1 to 3 ms [13,14]. In these areas, the inhibitory GABAergic currents have a rise time between 1 and 2 ms [15] and a decay time varying from about 5 ms to about 30 ms [16]. Also the membrane time constants of different cortical neurons vary over a wide range, from close to 5 ms to well over over 40 ms [17,18,19,20]. These physiological data show that the decay times for AMPA synapses are similar or larger than the GABA rise times and furthermore that the GABA decay times can be larger than the membrane time constant. However, as stated above the original treatment of the Carnevale-Hines integration scheme requires that the excitatory decay time should be smaller th
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