The neuronal signal consists of short voltage pulses called action potentials or spikes. These pulses travel along the axon and are distributed to several postsynaptic neurons where they evoke postsynaptic potentials. If a postsynaptic neuron receives a sufficient number of spikes from several presynaptic neurons within a short time window, its membrane potential may reach a critical value and an action potential is triggered. We say that the neuron has ‘fired’ a spike. This spike is the neuronal output signal which is, in turn, transmitted to other neurons.
A particularly simple model of a spiking neuron is the leaky integrate-and-fire model. First, a linear differential equation describes how input currents are integrated and transformed into a membrane voltage $u(t)$. Here the input can be the input current injected by an experimentalist into an isolated neuron or synaptic input currents caused by spikes arriving from other neurons in a large and highly connected network. Second, the model neuron generates an output spike if the membrane voltage reaches the threshold $\vartheta$. Finally, after spike firing, the integration of the linear differential equation resumes from a reset value $u_{r}$.
The simple leaky integrate-and-fire model does not account for long-lasting refractoriness or adaptation. However, if the voltage dynamics of the leaky integrate-and-fire model is enhanced by mechanisms of adaptation, then it can be a powerful tool to accurately predict spike times of cortical neurons. Such generalized integrate-and-fire models are the main topic of Part II of this book.
An elementary, non-technical introduction to neurons and synapses can be found in the book by Thompson (511). At an intermediate level is the introductory textbook of Purves et al. (409) while the “Principles of Neural Science” by Kandel et al. (249) can be considered as a standard textbook on neuroscience covering a wealth of experimental results.
The use of mathematics to explain neuronal activity has a long tradition in theoretical neuroscience, over one hundred years. Phenomenological spiking neuron models similar to the leaky integrate-and-fire model have been proposed by Lapique in 1907 who wanted to predict the first spike after stimulus onset (so that his model did not yet have the reset of the membrane potential after firing), and have been developed further in different variants by others (288; 219; 335; 494; 170; 547; 495). For the ‘filter’ description of integrate-and-fire models see for example Gerstner et al. (1996) and Pillow et al. (1998). The elegance and simplicity of integrate-and-fire models makes them a widely used tool to describe principles of neural information processing in neural networks of a broad range of sizes.
A different line of mathematical neuron models are biophysical models, first developed by Hodgkin and Huxley (222); these biophysical models are the topic of the next chapter.
Synaptic current pulse . Synaptic inputs can be approximated by an exponential current $I(t)=q\,{1\over\tau_{s}}\exp[-{t-t^{(f)}\over\tau_{s}}]$ for $t>t^{(f)}$ where $t^{(f)}$ is the moment when the spike arrives at the synapse.
(a) Use Eq. ( 1.5 ) to calculate the response of a passive membrane with time constant $\tau_{m}$ to an input spike arriving at time $t^{(f)}$ .
(b) In the solution resulting from (a), take the limit $\tau_{s}\to\tau_{m}$ and show that in this limit the response is proportional to $\propto[t-t^{(f)}]\,\exp[-{t-t^{(f)}\over\tau_{s}}]$ . A function of this form is sometimes called an $\alpha$ -function.
(c) In the solution resulting from (a), take the limit $\tau_{s}\to 0$ . Can you relate your result to the discussion of the Dirac- $\delta$ function?
Time-dependent solution . Show that Eq. ( 1.15 ) is a solution of the differential equation Eq. ( 1.5 ) for time-dependent input $I(t)$ . To do so, start by changing the variable in the integral from $s$ to $t^{\prime}=t-s$ . Then take the derivative of Eq. ( 1.15 ) and compare the terms to those on both sides of the differential equation.
Chain of linear equations. Suppose that arrival of a spike at time $t^{(f)}$ releases neurotransmitter into the synaptic cleft. The amount of available neurotransmitter at time $t$ is $\tau_{x}\,{{\text{d}}x\over{\text{d}}t}=-x+\delta(t-t^{(f)})\,.$ The neurotransmitter binds to the postsynaptic membrane and opens channels that enable a synaptic current $\tau_{s}\,{{\text{d}}I\over{\text{d}}t}=-I+I_{0}\,x(t)\,.$ Finally, the current charges the postsynaptic membrane according to $\tau_{m}\,{{\text{d}}u\over{\text{d}}t}=-u+R\,I(t).$ Write the voltage response to a single current pulse as an integral.
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