Imbalanced amplification: A mechanism of amplification and suppression from local imbalance of excitation and inhibition in cortical circuits

<div><p>Understanding the relationship between external stimuli and the spiking activity of cortical populations is a central problem in neuroscience. Dense recurrent connectivity in local cortical circuits can lead to counterintuitive response properties, raising the question of whether there are simple arithmetical rules for relating circuits’ connectivity structure to their response properties. One such arithmetic is provided by the mean field theory of balanced networks, which is derived in a limit where excitatory and inhibitory synaptic currents precisely balance on average. However, balanced network theory is not applicable to some biologically relevant connectivity structures. We show that cortical circuits with such structure are susceptible to an amplification mechanism arising when excitatory-inhibitory balance is broken at the level of local subpopulations, but maintained at a global level. This amplification, which can be quantified by a linear correction to the classical mean field theory of balanced networks, explains several response properties observed in cortical recordings and provides fundamental insights into the relationship between connectivity structure and neural responses in cortical circuits.</p></div

Imprecise balance under optogenetic stimulation.

<p><b>a)</b> Schematic. A population of recurrently connected excitatory (red) and inhibitory (blue) spiking neuron models receive synaptic input from an external population (<i>X</i>; green) of Poisson-spiking neurons. Optogenetic stimulation of excitatory neurons was modeled by an extra inward current to the excitatory population at 5s. <b>b)</b> Spike rasters from 50 randomly selected excitatory (red) and inhibitory (blue) neurons from recurrent network. <b>c)</b> Average firing rate of excitatory (red) and inhibitory (blue) neurons in the recurrent network from simulations (light solid), from the balanced network approximation (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006048#pcbi.1006048.e008" target="_blank">Eq (3)</a>; solid dark) and from the corrected approximation (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006048#pcbi.1006048.e010" target="_blank">Eq (4)</a>; dashed). <b>d)</b> Mean synaptic currents to 200 randomly selected excitatory neurons in the recurrent network from external inputs (<i>X</i>; green), from the local population (<i>E</i> + <i>I</i>; purple) and the total synaptic current (black). Currents are measured in units of the neurons’ rheobase (rheobase/<i>C</i>_<i>m</i> = 10.5 V/s). <b>e)</b> Mean firing rates plotted against mean input currents to all neurons in populations <i>E</i> and <i>I</i> (gray dots) and a rectified linear fit to their relationship (black line). <b>f)</b> Mean firing rates from identical simulations without stimulation except the total number of neurons, <i>N</i>, in the recurrent network was modulated while scaling synaptic weights and connection probabilities so that (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006048#sec020" target="_blank">Methods</a>). Solid light curves are from simulations, solid dark from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006048#pcbi.1006048.e008" target="_blank">Eq (3)</a>, and dashed from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006048#pcbi.1006048.e010" target="_blank">Eq (4)</a>.</p

Imbalanced amplification and suppression reverse the blurring introduced by interlaminar synaptic divergence.

<p><b>a)</b> A disc-shaped grating stimulus gives rise to <b>b)</b> a disc-shaped firing rate profile, <i>r</i>_<i>X</i>(<b>x</b>), in L4 with slightly blurred edges (achieved by convolving contrast from a with a Gaussian kernel). <b>c)</b> Input, <i>X</i>_<i>E</i>(<b>x</b>), from L4 to excitatory neurons in L2/3 is blurred by synaptic divergence, which effectively applies a low-pass filter, , to the L4 rates. <b>d)</b> Excitatory firing rates in L2/3 are sharper than external input when lateral excitation is similar, but smaller, in width than interlaminar excitation (<i>α</i>_<i>E</i> = 0.85<i>α</i>_<i>X</i>). <b>e)</b> Same as c, but lateral excitation is exactly as broad as interlaminar excitation (<i>α</i>_<i>E</i> = <i>α</i>_<i>X</i>), which sharpens the edges further, making firing rates in L2/3 similar to those in L4. <b>f)</b> Same as c, but lateral excitation is broader than interlaminar excitation (<i>α</i>_<i>E</i> = 1.1<i>α</i>_<i>X</i>), which sharpens the edges even further, but also introduced suppressed regions due to Gibbs phenomena. <b>g-l)</b> Same as a-f, but contrast was determined by the brightness of a photograph. Horizontal and vertical axes are neurons’ receptive fields.</p

Imbalanced amplification and suppression under partial optogenetic stimulation.

<p>Same as Fig. 2a-d except the inward current was only provided to 20% of the excitatory neurons, modeling ChR2-expressing pyramidal cells. <b>a)</b> Schematic. A population of recurrently connected excitatory (red) and inhibitory (blue) spiking neuron models receive synaptic input from an external population (<i>X</i>; green) of Poisson-spiking neurons. Optogenetic stimulation of excitatory neurons was modeled by an extra inward current to 20% of the excitatory population at 5s. <b>b)</b> Spike rasters from 10 randomly selected ChR2-expressing and 40 non-expressing excitatory (red) neurons and 50 inhibitory (blue) neurons from the recurrent network. <b>c)</b> Average firing rate of ChR2-expressing excitatory neurons from simulations (light solid) and from the corrected approximation (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006048#pcbi.1006048.e010" target="_blank">Eq (4)</a>; dashed). <b>d)</b> Mean synaptic currents to 200 randomly selected ChR2-expressing excitatory neurons from external inputs (<i>X</i>; green), from the local population (<i>E</i> + <i>I</i>; purple) and the total synaptic current (black). <b>e)</b> Same as c, but for non-expressing excitatory neurons (red) and inhibitory neurons (blue). <b>f)</b> Same as d, but for non-expressing excitatory postsynaptic neurons. <b>g)</b> Same as c and d, but averaged over all excitatory neurons (expressing and non-expressing). Currents are measured in units of the neurons’ rheobase (rheobase/<i>C</i>_<i>m</i> = 10.5 V/s). Firing rates predicted by <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006048#pcbi.1006048.e008" target="_blank">Eq (3)</a> are not shown in c and e because <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006048#pcbi.1006048.e008" target="_blank">Eq (3)</a> is not applicable to those cases.</p

Response properties of a continuously indexed network.

<p><b>a)</b> Network diagram. Poisson spiking neurons in L4 (<i>X</i>) provide external synaptic input to 2 × 10⁵ recurrently connected excitatory and inhibitory AdEx model neurons (<i>E</i> and <i>I</i>) in L2/3. The spatial width of synaptic projections from population <i>a</i> = <i>X</i>, <i>E</i>, <i>I</i> is given by <i>β</i>_<i>a</i>. <b>b)</b> Neurons are assigned random orientations and connection probability also depends on the difference, <i>dθ</i>, between neurons’ preferred orientation. <b>c)</b> An oriented stimulus in the animal’s visual field. <b>d,e)</b> The location of the stimulus is modeled by firing rates in L4 that are peaked at the location of the stimulus in physical and orientation space. <b>f,g)</b> Synaptic current to neurons in population <i>E</i> from the external network (green), the local network (purple) and total (black) as a function distance from the receptive field center and as a function of neurons’ preferred orientation. <b>h,i)</b> Firing rate profiles of excitatory (red) and inhibitory (blue) neurons in the local network from simulations (light curves), classical balanced network theory (solid, dark curves; from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006048#pcbi.1006048.e073" target="_blank">Eq (13)</a>) and under the linear correction (dashed; from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006048#pcbi.1006048.e089" target="_blank">Eq (17)</a>) in physical and orientation space. <b>j-o)</b> Same as (d-i) except for a smaller visual stimulus, modeled by a narrower spatial firing rate profile in L4. Firing rates from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006048#pcbi.1006048.e073" target="_blank">Eq (13)</a> are not shown in panels n and o because balance cannot be realized and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006048#pcbi.1006048.e073" target="_blank">Eq (13)</a> cannot be applied when external input is narrower than recurrent connectivity (see main text, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006048#pcbi.1006048.s001" target="_blank">S1 Text</a>, and [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006048#pcbi.1006048.ref008" target="_blank">8</a>]).</p

Imbalanced amplification of weak stimuli.

<p>Same simulations as Figs <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006048#pcbi.1006048.g002" target="_blank">2</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006048#pcbi.1006048.g003" target="_blank">3</a> except with <i>N</i> increased by a factor of four, <i>ϵ</i> decreased by a factor of two, and a weaker stimulus. <b>a)</b> Schematic. A recurrent network (stimulated layer), which receives external input from Poisson-spiking neurons (green X) and from partial optogenetic stimulation, sends excitatory synaptic input to an identical network (downstream layer). <b>b)</b> Average firing rate of ChR2-expressing excitatory neurons in the stimulated layer from simulations (light solid) and from the corrected approximation (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006048#pcbi.1006048.e010" target="_blank">Eq (4)</a>; dashed). <b>c)</b> Same as b, but for non-expressing excitatory (red) and inhibitory (blue) neurons in the stimulated layer. <b>d)</b> Same as b, but averaged over all excitatory neurons in the stimulated layer. <b>e)</b> Same as d, but for the downstream layer. Mean firing rates from simulations of the stimulated layer changed from 5.8 Hz before stimulation to 10.0 Hz during stimulation for ChR2-expressing neurons, from 5.9 to 5.1 Hz for non-expressing excitatory neurons, from 5.9 to 6.1 Hz averaged over all excitatory neurons, and from 7.8 to 8.0 Hz for inhibitory neurons. Mean firing rates from simulations of the downstream layer changed from 7.2 Hz to 8.1 Hz for excitatory neurons and from 8.5 Hz to 9.5 Hz for inhibitory neurons.</p

Data for "Imbalanced Amplification"

Data and Matlab code for reproducing figures in "Imbalanced amplification: A mechanism of amplification and suppression from local imbalance of excitation and inhibition in cortical circuits"<br><br