# The Balanced Network

Toy example of a random graph generated with the networkx python package

## Background

When we have a satisfactory model for one neuron (see Integrate-and-fire section), we can think about modelling a network of neurons. By connecting them with synapses, network models are a powerful tool to understand brain dynamics and the origin of the electrical activity observed in vivo. The type of the neuron models used and the connectivity scheme influence the kind of dynamics that can be observed: in the following, we will focus mainly on recurrent networks of integrate and fire neurons. Among alternatives approaches, networks of binary neurons are easier to analyse analytically [Vreeswijk1996, Vreeswijk1998, Sompolinsky1988]. Those models treat the neuron like the spin of an elementary particle, its activity being a binary variable (spiking or silent), and established a link between theoretical physic and neuroscience. Models such as the Ising model [Schneidman2006, Marre2009] are used to infer correlations and structure of the neuronal code. Behind all these models, the key point is to explain the irregularity of the neuronal discharges observed in vivo. To explain it without stochastic inputs, one needs to have large fluctuations of the neuronal dynamics, counterbalanced by weak synaptic weights and by the fact that the activity should be balanced: an excitatory and and inhibitory population should act such that the average activity stays below a certain threshold, while fluctuations may be high and let the neurons cross a threshold, in order to emit spikes.

## The balanced network

The balanced random network [Vreeswijk1996, Vreeswijk1998, Brunel2000, Vogels2005a, Kumar2008a, ElBoustani2009a, Amit1997, Renart2010] is a common and convenient framework for studying the dynamics of large-scale populations of sparsely-connected integrate-and-fire neurons. In these networks, two generic populations of excitatory and inhibitory neurons are reciprocally coupled with weights  and  (see Figure) to generate a balanced regime where the average depolarization of the neurons is roughly constant, subthreshold, and irregular spiking is the result of fluctuations. There is a classical ratio of 4 excitatory neurons for 1 inhibitory neurons, based on the measured ratio in cortex  and a sparse, random connectivity. Every neuron is typically connected to  of the others, and depending on certain key parameters, mainly the amount of external noise injected into the system and the balance between excitatory and inhibitory weights, several regimes of activity can be observed. Those regime have been described and classified in (Brunel, 2000), and can be asynchronous/synchronous (from a population viewpoint) and regular/irregular (from a neuron viewpoint).

Left: Schema of a random balanced network, with two excitatory and inhibitory populations interacting together. Right: Spikes produced by all the neurons in such a network
The main activity regimes observed in neuronal networks. Top Left: Synchronous regular (SR): the global activity of the network is oscillatory, and all the neurons fire regularly at intervals of their refractory period. Top Right: Synchronous irregular (SI): the global activity is oscillatory, but neurons fire irregularly as Poisson-like sources. Bottom Left: Asynchronous regular (AR), the global activity is constant, but neurons fire regularly. Bottom Left: Asynchronous irregular (AI), both the individual discharges of the neurons and the global firing rate are irregular. Adapted from (Brunel, 2000)

## Classification of the dynamics

The average firing rate of all the neurons within the network can be constant (asynchronous) or display oscillations (synchronous). The individual discharge of one neuron can be regular (the inter spike intervals (ISIs) are almost all equal), or irregular (the ISIs follow a Poisson distribution). This irregularity is often quantified by the coefficient of variation (CV), given by  where  denotes the average and  the standard deviation. A pure Poisson process has a CV equal to 1. The more the discharge is regular, the more the CV tends to 0. Classical values observed in vivo in the spontaneous regime are usually close to 1, so simulations tend to focus on the irregular regime. In such a regime, neurons fire in an irregular manner, behaving almost like Poisson processes, and the average pairwise cross-correlation is modulated by the internal balance or the external input. This regime is also well suited to produce slow oscillations comparable with oscillations observed in vivo under anaesthesia [Han2008, Arieli1996]. Mean field models are a common tool used to establish and predict, analytically, the stationary average response of homogeneous networks of integrate-and-fire neurons under certain assumptions (neuronal discharges should be independent and Poissonian, in the irregular regime) [ElBoustani2009a, Brunel2000]. However, they are much harder to use when complex models of neurons are used, or when inhomogeneities, such as delays, are taken into account.

## Mean field models

While the integrate-and-fire and compartmental models consider that the exact times of spike occurrence are important and may play a role in the coding strategies used in the cortex, other models consider that the pertinent information is in the instantaneous firing rate of the neuron. Since the discharge of the cell can be noisy and irregular, the spikes are not modelled and the only relevant information used by those models is the firing rate of the neuron. At each time, the neuron can emit spikes with a certain probability , directly related to the activities  of its pre-synaptic sources, weighted by some factors :

\frac{dr(t)}{dt} = \sum_i w_i f(r_i(t))

where  is a positive, monotonic and increasing function, inducing a non linear relationship between the summed inputs and the instantaneous firing rate . Usually,  is a sigmoidal or a hyperbolic tangent function. An alternative viewpoint is to say that  represent the average firing rate over a population of identical neurons, rather than an instantaneous frequency, and that is why these equations are called mean-field or rate-based models. Solid mathematical results can be obtained with these mean field models, concerning either their dynamics or their learning properties.

Global schematic of a rate-based model. Each neuron is seen as a local average of a bunch of homogeneous neurons

## References

[Vreeswijk1996] C. van Vreeswijk and H. Sompolinsky, "Chaos in neuronal networks with balanced excitatory and inhibitory activity.," Science, vol. 274, pp. 1724-1726, 1996.
[Bibtex]
@article{Vreeswijk1996,
abstract = {Neurons in the cortex of behaving animals show temporally irregular spiking patterns. The origin of this irregularity and its implications for neural processing are unknown. The hypothesis that the temporal variability in the firing of a neuron results from an approximate balance between its excitatory and inhibitory inputs was investigated theoretically. Such a balance emerges naturally in large networks of excitatory and inhibitory neuronal populations that are sparsely connected by relatively strong synapses. The resulting state is characterized by strongly chaotic dynamics, even when the external inputs to the network are constant in time. Such a network exhibits a linear response, despite the highly nonlinear dynamics of single neurons, and reacts to changing external stimuli on time scales much smaller than the integration time constant of a single neuron.},
author = {van Vreeswijk, C and Sompolinsky, H},
journal = {Science},
keywords = {Animals; Cerebral Cortex; Haplorhini; Models,Neurological; Nerve Net; Neurons; Nonlinear Dynami},
pages = {1724--1726},
pmid = {8939866},
title = {{Chaos in neuronal networks with balanced excitatory and inhibitory activity.}},
volume = {274},
year = {1996}
}
[Vreeswijk1998] C. van Vreeswijk, H. Sompolinsky, and V. C. Vreeswijk, "Chaotic balanced state in a model of cortical circuits.," Neural computation, vol. 10, iss. 6, pp. 1321-71, 1998.
[Bibtex]
@article{Vreeswijk1998,
abstract = {The nature and origin of the temporal irregularity in the electrical activity of cortical neurons in vivo are not well understood. We consider the hypothesis that this irregularity is due to a balance of excitatory and inhibitory currents into the cortical cells. We study a network model with excitatory and inhibitory populations of simple binary units. The internal feedback is mediated by relatively large synaptic strengths, so that the magnitude of the total excitatory and inhibitory feedback is much larger than the neuronal threshold. The connectivity is random and sparse. The mean number of connections per unit is large, though small compared to the total number of cells in the network. The network also receives a large, temporally regular input from external sources. We present an analytical solution of the mean-field theory of this model, which is exact in the limit of large network size. This theory reveals a new cooperative stationary state of large networks, which we term a balanced state. In this state, a balance between the excitatory and inhibitory inputs emerges dynamically for a wide range of parameters, resulting in a net input whose temporal fluctuations are of the same order as its mean. The internal synaptic inputs act as a strong negative feedback, which linearizes the population responses to the external drive despite the strong nonlinearity of the individual cells. This feedback also greatly stabilizes the system's state and enables it to track a time-dependent input on time scales much shorter than the time constant of a single cell. The spatiotemporal statistics of the balanced state are calculated. It is shown that the autocorrelations decay on a short time scale, yielding an approximate Poissonian temporal statistics. The activity levels of single cells are broadly distributed, and their distribution exhibits a skewed shape with a long power-law tail. The chaotic nature of the balanced state is revealed by showing that the evolution of the microscopic state of the network is extremely sensitive to small deviations in its initial conditions. The balanced state generated by the sparse, strong connections is an asynchronous chaotic state. It is accompanied by weak spatial cross-correlations, the strength of which vanishes in the limit of large network size. This is in contrast to the synchronized chaotic states exhibited by more conventional network models with high connectivity of weak synapses.},
annote = { From Duplicate 1 ( Chaotic balanced state in a model of cortical circuits. - Vreeswijk, C Van; Sompolinsky, H; van Vreeswijk, C )
From Duplicate 1 ( Chaotic balanced state in a model of cortical circuits. - van Vreeswijk, C; Sompolinsky, H )
From Duplicate 2 ( Chaotic balanced state in a model of cortical circuits. - van Vreeswijk, C; Sompolinsky, H; Vreeswijk, C Van )
From Duplicate 1 ( Chaotic balanced state in a model of cortical circuits. - Vreeswijk, C Van; Sompolinsky, H; van Vreeswijk, C )
From Duplicate 1 ( Chaotic balanced state in a model of cortical circuits. - van Vreeswijk, C; Sompolinsky, H )
From Duplicate 2 ( Chaotic balanced state in a model of cortical circuits. - van Vreeswijk, C; Sompolinsky, H )
},
author = {van Vreeswijk, C and Sompolinsky, H and Vreeswijk, C Van},
file = {:home/pierre/Mendeley//van Vreeswijk, Sompolinsky, Vreeswijk - 1998.pdf:pdf},
issn = {0899-7667},
journal = {Neural computation},
keywords = {Animals,Cerebral Cortex,Cerebral Cortex: cytology,Cerebral Cortex: physiology,Models,Neurological,Neurons,Neurons: physiology,Nonlinear Dynamics,Sensory,Sensory Thresholds,Sensory Thresholds: physiology,Synapses,Synapses: physiology,Time Factors},
month = aug,
number = {6},
pages = {1321--71},
pmid = {9698348},
title = {{Chaotic balanced state in a model of cortical circuits.}},
url = {http://www.ncbi.nlm.nih.gov/pubmed/9698348},
volume = {10},
year = {1998}
}
[Sompolinsky1988] H. Sompolinsky, A. Crisanti, and H. J. Sommers, "Chaos in random neural networks," Physical review letters, vol. 61, iss. 3, pp. 259-262, 1988.
[Bibtex]
@article{Sompolinsky1988,
annote = {
From Duplicate 2 (
Chaos in random neural networks
- Sompolinsky, H; Crisanti, A; Sommers, HJ J )
From Duplicate 2 (
Chaos in random neural networks
- Sompolinsky, H; Crisanti, A; Sommers, HJ J )
},
author = {Sompolinsky, H and Crisanti, A and Sommers, HJ J},
file = {:home/pierre/Mendeley/Sompolinsky, Crisanti, Sommers - 1988.pdf:pdf;:home/pierre/Mendeley//Sompolinsky, Crisanti, Sommers - 1988.pdf:pdf},
journal = {Physical Review Letters},
number = {3},
pages = {259--262},
pmid = {10039285},
title = {{Chaos in random neural networks}},
volume = {61},
year = {1988}
}
[Schneidman2006] E. Schneidman, M. J. Berry, R. Segev, and W. Bialek, "Weak pairwise correlations imply strongly correlated network states in a neural population.," Nature, vol. 440, iss. 7087, pp. 1007-1012, 2006.
[Bibtex]
@article{Schneidman2006,
abstract = {Biological networks have so many possible states that exhaustive sampling is impossible. Successful analysis thus depends on simplifying hypotheses, but experiments on many systems hint that complicated, higher-order interactions among large groups of elements have an important role. Here we show, in the vertebrate retina, that weak correlations between pairs of neurons coexist with strongly collective behaviour in the responses of ten or more neurons. We find that this collective behaviour is described quantitatively by models that capture the observed pairwise correlations but assume no higher-order interactions. These maximum entropy models are equivalent to Ising models, and predict that larger networks are completely dominated by correlation effects. This suggests that the neural code has associative or error-correcting properties, and we provide preliminary evidence for such behaviour. As a first test for the generality of these ideas, we show that similar results are obtained from networks of cultured cortical neurons.},
author = {Schneidman, Elad and Berry, Michael J and Segev, Ronen and Bialek, William},
doi = {10.1038/nature04701},
file = {:home/pierre/Mendeley/Schneidman et al. - 2006.pdf:pdf},
issn = {1476-4687},
journal = {Nature},
keywords = {Action Potentials; Animals; Cerebral Cortex; Entro,Extramural; Research Support,N.I.H.,Neurological; Neurons; Poisson Distribution; Resea,Non-U.S. Gov't; Retina; Urodela},
number = {7087},
pages = {1007--1012},
pmid = {16625187},
shorttitle = {Nature},
title = {{Weak pairwise correlations imply strongly correlated network states in a neural population.}},
url = {http://dx.doi.org/10.1038/nature04701},
volume = {440},
year = {2006}
}
[Marre2009] O. Marre, P. Yger, A. P. Davison, and Y. Frégnac, "Reliable recall of spontaneous activity patterns in cortical networks.," The journal of neuroscience : the official journal of the society for neuroscience, vol. 29, iss. 46, pp. 14596-606, 2009.
[Bibtex]
@article{Marre2009,
abstract = {Irregular ongoing activity in cortical networks is often modeled as arising from recurrent connectivity. Yet it remains unclear to what extent its presence corrupts sensory signal transmission and network computational capabilities. In a recurrent cortical-like network, we have determined the activity patterns that are better transmitted and self-sustained by the network. We show that reproducible spiking and subthreshold dynamics can be triggered if the statistics of the imposed external drive are consistent with patterns previously seen in the ongoing activity. A subset of neurons in the network, constrained to replay temporal pattern segments extracted from the recorded ongoing activity of the same network, reliably drives the remaining, free-running neurons to call the rest of the pattern. Comparison with surrogate Poisson patterns indicates that the efficiency of the recall and completion process depends on the similarity between the statistical properties of the input with previous ongoing activity The reliability of evoked dynamics in recurrent networks is thus dependent on the stimulus used, and we propose that the similarity between spontaneous and evoked activity in sensory cortical areas could be a signature of efficient transmission and propagation across cortical networks.},
author = {Marre, Olivier and Yger, Pierre and Davison, Andrew P and Fr\'{e}gnac, Yves},
doi = {10.1523/JNEUROSCI.0753-09.2009},
file = {:home/pierre/Mendeley/Marre et al. - 2009.pdf:pdf},
issn = {1529-2401},
journal = {The Journal of neuroscience : the official journal of the Society for Neuroscience},
keywords = {Action Potentials,Action Potentials: physiology,Animals,Cerebral Cortex,Cerebral Cortex: physiology,Humans,Mental Recall,Mental Recall: physiology,Models, Neurological,Nerve Net,Nerve Net: physiology,Reproducibility of Results},
month = nov,
number = {46},
pages = {14596--606},
pmid = {19923292},
title = {{Reliable recall of spontaneous activity patterns in cortical networks.}},
url = {http://www.ncbi.nlm.nih.gov/pubmed/19923292},
volume = {29},
year = {2009}
}
[Brunel2000] N. Brunel, "Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons.," J comput neurosci, vol. 8, pp. 183-208, 2000.
[Bibtex]
@article{Brunel2000,
abstract = {The dynamics of networks of sparsely connected excitatory and inhibitory integrate-and-fire neurons are studied analytically. The analysis reveals a rich repertoire of states, including synchronous states in which neurons fire regularly; asynchronous states with stationary global activity and very irregular individual cell activity; and states in which the global activity oscillates but individual cells fire irregularly, typically at rates lower than the global oscillation frequency. The network can switch between these states, provided the external frequency, or the balance between excitation and inhibition, is varied. Two types of network oscillations are observed. In the fast oscillatory state, the network frequency is almost fully controlled by the synaptic time scale. In the slow oscillatory state, the network frequency depends mostly on the membrane time constant. Finite size effects in the asynchronous state are also discussed.},
author = {Brunel, N},
file = {:home/pierre/Mendeley/Brunel - 2000.pdf:pdf},
journal = {J Comput Neurosci},
keywords = {Action Potentials; Biological Clocks; Brain; Corti,Neurological; Nerve Net; Neural Inhibition; Neural},
pages = {183--208},
pmid = {10809012},
title = {{Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons.}},
volume = {8},
year = {2000}
}
[Vogels2005a] T. P. Vogels and L. F. Abbott, "Signal propagation and logic gating in networks of integrate-and-fire neurons.," J neurosci, vol. 25, iss. 46, pp. 10786-10795, 2005.
[Bibtex]
@article{Vogels2005a,
abstract = {Transmission of signals within the brain is essential for cognitive function, but it is not clear how neural circuits support reliable and accurate signal propagation over a sufficiently large dynamic range. Two modes of propagation have been studied: synfire chains, in which synchronous activity travels through feedforward layers of a neuronal network, and the propagation of fluctuations in firing rate across these layers. In both cases, a sufficient amount of noise, which was added to previous models from an external source, had to be included to support stable propagation. Sparse, randomly connected networks of spiking model neurons can generate chaotic patterns of activity. We investigate whether this activity, which is a more realistic noise source, is sufficient to allow for signal transmission. We find that, for rate-coded signals but not for synfire chains, such networks support robust and accurate signal reproduction through up to six layers if appropriate adjustments are made in synaptic strengths. We investigate the factors affecting transmission and show that multiple signals can propagate simultaneously along different pathways. Using this feature, we show how different types of logic gates can arise within the architecture of the random network through the strengthening of specific synapses.},
author = {Vogels, Tim P and Abbott, L F},
doi = {10.1523/JNEUROSCI.3508-05.2005},
file = {:home/pierre/Mendeley/Vogels, Abbott - 2005.pdf:pdf},
issn = {1529-2401},
journal = {J Neurosci},
keywords = {Action Potentials; Ion Channel Gating; Logic; Mode,Neurological; Neural Networks (Computer); Neural P},
month = nov,
number = {46},
pages = {10786--10795},
pmid = {16291952},
shorttitle = {J Neurosci},
title = {{Signal propagation and logic gating in networks of integrate-and-fire neurons.}},
url = {http://dx.doi.org/10.1523/JNEUROSCI.3508-05.2005},
volume = {25},
year = {2005}
}
[Kumar2008a] A. Kumar, S. Schrader, A. Aertsen, and S. Rotter, "The high-conductance state of cortical networks.," Neural computation, vol. 20, iss. 1, pp. 1-43, 2008.
[Bibtex]
@article{Kumar2008a,
abstract = {We studied the dynamics of large networks of spiking neurons with conductance-based (nonlinear) synapses and compared them to networks with current-based (linear) synapses. For systems with sparse and inhibition-dominated recurrent connectivity, weak external inputs induced asynchronous irregular firing at low rates. Membrane potentials fluctuated a few millivolts below threshold, and membrane conductances were increased by a factor 2 to 5 with respect to the resting state. This combination of parameters characterizes the ongoing spiking activity typically recorded in the cortex in vivo. Many aspects of the asynchronous irregular state in conductance-based networks could be sufficiently well characterized with a simple numerical mean field approach. In particular, it correctly predicted an intriguing property of conductance-based networks that does not appear to be shared by current-based models: they exhibit states of low-rate asynchronous irregular activity that persist for some period of time even in the absence of external inputs and without cortical pacemakers. Simulations of larger networks (up to 350,000 neurons) demonstrated that the survival time of self-sustained activity increases exponentially with network size.},
author = {Kumar, Arvind and Schrader, Sven and Aertsen, Ad and Rotter, Stefan},
doi = {10.1162/neco.2008.20.1.1},
file = {:home/pierre/Mendeley/Kumar et al. - 2008.pdf:pdf},
issn = {0899-7667},
journal = {Neural computation},
keywords = {Action Potentials,Action Potentials: physiology,Animals,Cell Membrane,Cell Membrane: physiology,Cerebral Cortex,Cerebral Cortex: physiology,Computer Simulation,Cortical Synchronization,Excitatory Postsynaptic Potentials,Excitatory Postsynaptic Potentials: physiology,Humans,Inhibitory Postsynaptic Potentials,Inhibitory Postsynaptic Potentials: physiology,Nerve Net,Nerve Net: physiology,Neural Networks (Computer),Neural Pathways,Neural Pathways: physiology,Neurons,Neurons: physiology,Nonlinear Dynamics,Synaptic Transmission,Synaptic Transmission: physiology,Time Factors},
month = jan,
number = {1},
pages = {1--43},
pmid = {18044999},
title = {{The high-conductance state of cortical networks.}},
url = {http://www.ncbi.nlm.nih.gov/pubmed/18044999},
volume = {20},
year = {2008}
}
[ElBoustani2009a] S. {El Boustani}, O. Marre, S. Béhuret, P. Baudot, P. Yger, T. Bal, A. Destexhe, and Y. Frégnac, "Network-state modulation of power-law frequency-scaling in visual cortical neurons.," Plos computational biology, vol. 5, iss. 9, p. e1000519, 2009.
[Bibtex]
@article{ElBoustani2009a,
abstract = {Various types of neural-based signals, such as EEG, local field potentials and intracellular synaptic potentials, integrate multiple sources of activity distributed across large assemblies. They have in common a power-law frequency-scaling structure at high frequencies, but it is still unclear whether this scaling property is dominated by intrinsic neuronal properties or by network activity. The latter case is particularly interesting because if frequency-scaling reflects the network state it could be used to characterize the functional impact of the connectivity. In intracellularly recorded neurons of cat primary visual cortex in vivo, the power spectral density of V(m) activity displays a power-law structure at high frequencies with a fractional scaling exponent. We show that this exponent is not constant, but depends on the visual statistics used to drive the network. To investigate the determinants of this frequency-scaling, we considered a generic recurrent model of cortex receiving a retinotopically organized external input. Similarly to the in vivo case, our in computo simulations show that the scaling exponent reflects the correlation level imposed in the input. This systematic dependence was also replicated at the single cell level, by controlling independently, in a parametric way, the strength and the temporal decay of the pairwise correlation between presynaptic inputs. This last model was implemented in vitro by imposing the correlation control in artificial presynaptic spike trains through dynamic-clamp techniques. These in vitro manipulations induced a modulation of the scaling exponent, similar to that observed in vivo and predicted in computo. We conclude that the frequency-scaling exponent of the V(m) reflects stimulus-driven correlations in the cortical network activity. Therefore, we propose that the scaling exponent could be used to read-out the "effective" connectivity responsible for the dynamical signature of the population signals measured at different integration levels, from Vm to LFP, EEG and fMRI.},
author = {{El Boustani}, Sami and Marre, Olivier and B\'{e}huret, S\'{e}bastien and Baudot, Pierre and Yger, Pierre and Bal, Thierry and Destexhe, Alain and Fr\'{e}gnac, Yves},
doi = {10.1371/journal.pcbi.1000519},
file = {:home/pierre/Mendeley/El Boustani et al. - 2009.pdf:pdf},
issn = {1553-7358},
journal = {PLoS computational biology},
keywords = {Animals,Cats,Computational Biology,Computational Biology: methods,Computer Simulation,Eye Movements,Eye Movements: physiology,Fractals,Membrane Potentials,Models, Neurological,Neurons,Neurons: physiology,Patch-Clamp Techniques,Photic Stimulation,Rats,Rats, Wistar,Visual Cortex,Visual Cortex: cytology,Visual Cortex: physiology},
month = sep,
number = {9},
pages = {e1000519},
pmid = {19779556},
title = {{Network-state modulation of power-law frequency-scaling in visual cortical neurons.}},
url = {http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2740863\&tool=pmcentrez\&rendertype=abstract},
volume = {5},
year = {2009}
}
[Amit1997] D. J. Amit and N. Brunel, "Model of global spontaneous activity and local structured activity during delay periods in the cerebral cortex.," Cerebral cortex (new york, n.y. : 1991), vol. 7, iss. 3, pp. 237-52.
[Bibtex]
@article{Amit1997,
abstract = {We investigate self-sustaining stable states (attractors) in networks of integrate-and-fire neurons. First, we study the stability of spontaneous activity in an unstructured network. It is shown that the stochastic background activity, of 1-5 spikes/s, is unstable if all neurons are excitatory. On the other hand, spontaneous activity becomes self-stabilizing in presence of local inhibition, given reasonable values of the parameters of the network. Second, in a network sustaining physiological spontaneous rates, we study the effect of learning in a local module, expressed in synaptic modifications in specific populations of synapses. We find that if the average synaptic potentiation (LTP) is too low, no stimulus specific activity manifests itself in the delay period. Instead, following the presentation and removal of any stimulus there is, in the local module, a delay activity in which all neurons selective (responding visually) to any of the stimuli presented for learning have rates which gradually increase with the amplitude of synaptic potentiation. When the average LTP increases beyond a critical value, specific local attractors (stable states) appear abruptly against the background of the global uniform spontaneous attractor. In this case the local module has two available types of collective delay activity: if the stimulus is unfamiliar, the activity is spontaneous; if it is similar to a learned stimulus, delay activity is selective. These new attractors reflect the synaptic structure developed during learning. In each of them a small population of neurons have elevated rates, which depend on the strength of LTP. The remaining neurons of the module have their activity at spontaneous rates. The predictions made in this paper could be checked by single unit recordings in delayed response experiments.},
author = {Amit, D J and Brunel, N},
issn = {1047-3211},
journal = {Cerebral cortex (New York, N.Y. : 1991)},
keywords = {Afferent,Afferent: physiology,Cerebral Cortex,Cerebral Cortex: cytology,Cerebral Cortex: physiology,Electrophysiology,Feedback,Feedback: physiology,Long-Term Potentiation,Long-Term Potentiation: physiology,Models,Motor Activity,Motor Activity: physiology,Neural Networks (Computer),Neurological,Neurons,Neurons: physiology,Poisson Distribution,Synaptic Membranes,Synaptic Membranes: physiology},
number = {3},
pages = {237--52},
pmid = {9143444},
title = {{Model of global spontaneous activity and local structured activity during delay periods in the cerebral cortex.}},
url = {http://www.ncbi.nlm.nih.gov/pubmed/9143444},
volume = {7}
}
[Renart2010] A. Renart, J. de la Rocha, P. Bartho, L. Hollender, N. Parga, A. Reyes, and K. D. Harris, "The asynchronous state in cortical circuits.," Science (new york, n.y.), vol. 327, iss. 5965, pp. 587-90, 2010.
[Bibtex]
@article{Renart2010,
abstract = {Correlated spiking is often observed in cortical circuits, but its functional role is controversial. It is believed that correlations are a consequence of shared inputs between nearby neurons and could severely constrain information decoding. Here we show theoretically that recurrent neural networks can generate an asynchronous state characterized by arbitrarily low mean spiking correlations despite substantial amounts of shared input. In this state, spontaneous fluctuations in the activity of excitatory and inhibitory populations accurately track each other, generating negative correlations in synaptic currents which cancel the effect of shared input. Near-zero mean correlations were seen experimentally in recordings from rodent neocortex in vivo. Our results suggest a reexamination of the sources underlying observed correlations and their functional consequences for information processing.},
author = {Renart, Alfonso and de la Rocha, Jaime and Bartho, Peter and Hollender, Liad and Parga, N\'{e}stor and Reyes, Alex and Harris, Kenneth D},
doi = {10.1126/science.1179850},
file = {:home/pierre/Mendeley/Renart et al. - 2010.pdf:pdf},
issn = {1095-9203},
journal = {Science (New York, N.Y.)},
keywords = {Action Potentials,Algorithms,Animals,Cerebral Cortex,Cerebral Cortex: cytology,Cerebral Cortex: physiology,Computer Simulation,Excitatory Postsynaptic Potentials,Inhibitory Postsynaptic Potentials,Models, Neurological,Nerve Net,Nerve Net: physiology,Neural Inhibition,Neural Pathways,Neural Pathways: physiology,Neurons,Neurons: physiology,Rats,Rats, Sprague-Dawley,Synapses,Synapses: physiology,Synaptic Potentials,Synaptic Transmission},
month = jan,
number = {5965},
pages = {587--90},
pmid = {20110507},
title = {{The asynchronous state in cortical circuits.}},
url = {http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2861483\&tool=pmcentrez\&rendertype=abstract},
volume = {327},
year = {2010}
}
[Han2008] F. Han, N. Caporale, and Y. Dan, "Reverberation of recent visual experience in spontaneous cortical waves.," Neuron, vol. 60, iss. 2, pp. 321-327, 2008.
[Bibtex]
@article{Han2008,
abstract = {Spontaneous waves of activity propagating across large cortical areas may play important roles in sensory processing and circuit refinement. However, whether these waves are in turn shaped by sensory experience remains unclear. Here we report that visually evoked cortical activity reverberates in subsequent spontaneous waves. Voltage-sensitive dye imaging in rat visual cortex shows that following repetitive presentation of a given visual stimulus, spatiotemporal activity patterns resembling the evoked response appear more frequently in the spontaneous waves. This effect is specific to the response pattern evoked by the repeated stimulus, and it persists for several minutes without further visual stimulation. Such wave-mediated reverberation could contribute to short-term memory and help to consolidate the transient effects of recent sensory experience into long-lasting cortical modifications.},
author = {Han, Feng and Caporale, Natalia and Dan, Yang},
doi = {10.1016/j.neuron.2008.08.026},
file = {:home/pierre/Mendeley/Han, Caporale, Dan - 2008.pdf:pdf},
institution = {Group in Vision Science, University of California Berkeley, Berkeley, CA 94720, USA.},
issn = {1097-4199},
journal = {Neuron},
keywords = {Action Potentials,Long-Evans; Staining and Labeling; Visual Cortex,Short-Term,Visual,ph/ysiology; Nerve Net,physiology,physiology; Animals; Evoked Potentials,physiology; Evoked Potentials,physiology; Fluorescent Dyes; Learning,physiology; Memory,physiology; Neurons,physiology; Photic Stimulation; Rats; Rats,physiology; Visual Pathways,physiology; Visual Perception},
month = oct,
number = {2},
pages = {321--327},
pmid = {18957223},
shorttitle = {Neuron},
title = {{Reverberation of recent visual experience in spontaneous cortical waves.}},
url = {http://dx.doi.org/10.1016/j.neuron.2008.08.026},
volume = {60},
year = {2008}
}
[Arieli1996] A. Arieli, A. Sterkin, A. Grinvald, and A. Aertsen, "Dynamics of ongoing activity: explanation of the large variability in evoked cortical responses.," Science, vol. 273, pp. 1868-1871, 1996.
[Bibtex]
@article{Arieli1996,
abstract = {Evoked activity in the mammalian cortex and the resulting behavioral responses exhibit a large variability to repeated presentations of the same stimulus. This study examined whether the variability can be attributed to ongoing activity. Ongoing and evoked spatiotemporal activity patterns in the cat visual cortex were measured with real-time optical imaging; local field potentials and discharges of single neurons were recorded simultaneously, by electrophysiological techniques. The evoked activity appeared deterministic, and the variability resulted from the dynamics of ongoing activity, presumably reflecting the instantaneous state of cortical networks. In spite of the large variability, evoked responses in single trials could be predicted by linear summation of the deterministic response and the preceding ongoing activity. Ongoing activity must play an important role in cortical function and cannot be ignored in exploration of cognitive processes.},
author = {Arieli, A and Sterkin, A and Grinvald, A and Aertsen, A},
file = {:home/pierre/Mendeley/Arieli et al. - 1996.pdf:pdf},
journal = {Science},
keywords = {Animals; Cats; Evoked Potentials,Computer-Assisted; Visual Cortex; Visual Pathways,Visual; Membrane Potentials; Neurons; Photic Stimu},
month = sep,
pages = {1868--1871},
pmid = {8791593},
title = {{Dynamics of ongoing activity: explanation of the large variability in evoked cortical responses.}},
volume = {273},
year = {1996}
}