A major goal of neuroscience, statistical physics and nonlinear dynamics is to understand how brain function arises from the collective dynamics of networks of spiking neurons. This challenge has been chiefly addressed through large-scale numerical simulations. Alternatively, researchers have formulated mean-field theories to gain insight into macroscopic states of large neuronal networks in terms of the collective firing activity of the neurons, or the firing rate. However, these theories have not succeeded in establishing an exact correspondence between the firing rate of the network and the underlying microscopic state of the spiking neurons. This has largely constrained the range of applicability of such macroscopic descriptions, particularly when trying to describe neuronal synchronization. Here we provide the derivation of a set of exact macroscopic equations for a network of spiking neurons. Our results reveal that the spike generation mechanism of individual neurons introduces an effective coupling between two biophysically relevant macroscopic quantities, the firing rate and the mean membrane potential, which together govern the evolution of the neuronal network. The resulting equations exactly describe all possible macroscopic dynamical states of the network, including states of synchronous spiking activity. Finally we show that the firing rate description is related, via a conformal map, with a lowdimensional description in terms of the Kuramoto order parameter, called Ott-Antonsen theory. We anticipate our results will be an important tool in investigating how large networks of spiking neurons self-organize in time to process and encode information in the brain.Processing and coding of information in the brain necessarily imply the coordinated activity of large ensembles of neurons. Within sensory regions of the cortex, many cells show similar responses to a given stimulus, indicating a high degree of neuronal redundancy at the local level. This suggests that information is encoded in the population response and hence can be captured via macroscopic measures of the network activity [1]. Moreover, the collective behavior of large neuronal networks is particularly relevant given that current brain measurement techniques, such as electroencephalography (EEG) or functional magnetic resonance imaging (fMRI), provide data which is necessarily averaged over the activity of a large number of neurons.The macroscopic dynamics of neuronal ensembles has been extensively studied through computational models of large networks of recurrently coupled spiking neurons, including Hodgkin-Huxley-type conductance-based neurons [2] as well as simplified neuron models, see e.g. [3][4][5]. In parallel, researchers have sought to develop statistical descriptions of neuronal networks, mainly in terms of a macroscopic observable that measures the mean rate at which neurons emit spikes, the firing rate [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. These descriptions, called firing-rate equations (FREs), have ...

The perforant-path projection to the hippocampus forms synapses in the apical tuft of CA1 pyramidal neurons. We used computer modeling to examine the function of these distal synaptic inputs, which led to three predictions that we confirmed in experiments using rat hippocampal slices. First, activation of CA1 neurons by the perforant path is limited, a result of the long distance between these inputs and the soma. Second, activation of CA1 neurons by the perforant path depends on the generation of dendritic spikes. Third, the forward propagation of these spikes is unreliable, but can be facilitated by modest activation of Schaffer-collateral synapses in the upper apical dendrites. This 'gating' of dendritic spike propagation may be an important activation mode of CA1 pyramidal neurons, and its modulation by neurotransmitters or long-term, activity-dependent plasticity may be an important feature of dendritic integration during mnemonic processing in the hippocampus.

The distribution of in vivo average firing rates within local cortical networks has been reported to be highly skewed and long tailed. The distribution of average single-cell inputs, conversely, is expected to be Gaussian by the central limit theorem. This raises the issue of how a skewed distribution of firing rates might result from a symmetric distribution of inputs. We argue that skewed rate distributions are a signature of the nonlinearity of the in vivo f-I curve. During in vivo conditions, ongoing synaptic activity produces significant fluctuations in the membrane potential of neurons, resulting in an expansive nonlinearity of the f-I curve for low and moderate inputs. Here, we investigate the effects of single-cell and network parameters on the shape of the f-I curve and, by extension, on the distribution of firing rates in randomly connected networks.