Neurons in the brain communicate by short electrical pulses, the so-called action potentials or spikes. How can we understand the process of spike generation? How can we understand information transmission by neurons? What happens if thousands of neurons are coupled together in a seemingly random network? How does the network connectivity determine the activity patterns? And, vice versa, how does the spike activity influence the connectivity pattern? These questions are addressed in this 2002 introduction to spiking neurons aimed at those taking courses in computational neuroscience, theoretical biology, biophysics, or neural networks. The approach will suit students of physics, mathematics, or computer science; it will also be useful for biologists who are interested in mathematical modelling. The text is enhanced by many worked examples and illustrations. There are no mathematical prerequisites beyond what the audience would meet as undergraduates: more advanced techniques are introduced in an elementary, concrete fashion when needed.

We introduce a two-dimensional integrate-and-fire model that combines an exponential spike mechanism with an adaptation equation, based on recent theoretical findings. We describe a systematic method to estimate its parameters with simple electrophysiological protocols (current-clamp injection of pulses and ramps) and apply it to a detailed conductance-based model of a regular spiking neuron. Our simple model predicts correctly the timing of 96% of the spikes (+/-2 ms) of the detailed model in response to injection of noisy synaptic conductances. The model is especially reliable in high-conductance states, typical of cortical activity in vivo, in which intrinsic conductances were found to have a reduced role in shaping spike trains. These results are promising because this simple model has enough expressive power to reproduce qualitatively several electrophysiological classes described in vitro.

."A Hebbian form of synaptic plasticity at inhibitory synapses generates balanced input currents and sparse neuronal responses that stabilize memory traces in neuronal networks" Cortical neurons receive balanced excitatory and inhibitory membrane currents.Here, we show that such a balance can be established and maintained in an experiencedependent manner by synaptic plasticity at inhibitory synapses. The mechanism we put forward provides an explanation for the sparse firing patterns observed in response to natural stimuli and fits well with a recently observed interaction of excitatory and inhibitory receptive field plasticity. We show that the introduction of inhibitory plasticity in suitable recurrent networks provides a homeostatic mechanism that leads to asynchronous irregular network states. Further, it can accommodate synaptic memories with activity patterns that become indiscernible from the background state, but can be re-activated by external stimuli. Our results suggest an essential role of inhibitory plasticity in the formation and maintenance of functional cortical circuitry. 1The balance of excitatory and inhibitory membrane currents a neuron experiences during stimulated and ongoing activity has been the topic of many recent studies (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14). This balance, first defined as equal average amounts of de-and hyperpolarizing membrane currents (from hereon referred to as "global balance") is thought to be essential for maintaining stability of cortical networks (1, 2). In the balanced state networks display asynchronous irregular (AI) dynamics that mimic activity patterns observed in cortical neurons. Such asynchronous network states facilitate rapid responses to small changes in the input (2-4), providing an ideal substrate for cortical signal processing (5,15,16). Pathologies that disrupt the balance of excitation and inhibition have often been implicated in neurological diseases such as epilepsy or schizophrenia (17, 18).Moreover, the input currents to a given cortical neuron are not merely globally balanced. Excitatory and inhibitory inputs are coupled also in time (6-8) and co-tuned for different stimulus features (9,10). The tight coupling of excitation and inhibition suggests a more precise, detailed balance, in which each excitatory input arrives at the cell together with an inhibitory counterpart, supposedly supplied through feedforward inhibition (Fig. 1 A). These observations fit well with models of cortical processing in which balanced sensory inputs are left unattended, but can be transiently (11), or persistently turned on by targeted disruptions of the balance (12-14).Although it is widely thought that the excitatory-inhibitory balance plays an important role for stability and information processing in cortical networks, it is still not understood by which mechanisms this balance is established and maintained in the presence of ongoing sensory experiences. Inspired by recent experimental results (9), we investigate the hypothesis that synaptic plastic...

Electrophysiological connectivity patterns in cortex often show a few strong connections, sometimes bidirectional, in a sea of weak connections. In order to explain these connectivity patterns, we use a model of Spike--Timing--Dependent Plasticity where synaptic changes depend on presynaptic spike arrival and the postsynaptic membrane potential, filtered with two different time constants. The model describes several nonlinear effects in STDP experiments, as well as the voltage dependence of plasticity. We show that in a simulated recurrent network of spiking neurons our plasticity rule leads not only to development of localized receptive fields, but also to connectivity patterns that reflect the neural code: for temporal coding paradigms with spatio--temporal input correlations, strong connections are predominantly unidirectional, whereas they are bidirectional under rate coded input with spatial correlations only. Thus variable connectivity patterns in the brain could reflect different coding principles across brain areas; moreover simulations suggest that plasticity is surprisingly fast.

Populations of neurons in motor cortex engage in complex transient dynamics of large amplitude during the execution of limb movements. Traditional network models with stochastically assigned synapses cannot reproduce this behavior. Here we introduce a class of cortical architectures with strong and random excitatory recurrence that is stabilized by intricate, fine-tuned inhibition, optimized from a control theory perspective. Such networks transiently amplify specific activity states and can be used to reliably execute multidimensional movement patterns. Similar to the experimental observations, these transients must be preceded by a steady-state initialization phase from which the network relaxes back into the background state by way of complex internal dynamics. In our networks, excitation and inhibition are as tightly balanced as recently reported in experiments across several brain areas, suggesting inhibitory control of complex excitatory recurrence as a generic organizational principle in cortex.