Abstracts of the 2014 Brains, Minds, and Machines Summer School

A compilation of abstracts from the student projects of the 2014 Brains, Minds, and Machines Summer School, held at Woods Hole Marine Biological Lab, May 29 - June 12, 2014.This work was supported by the Center for Brains, Minds and Machines (CBMM), funded by NSF STC award CCF-1231216

Stability and response times in balanced networks

Persistent neuronal activity is usually studied in the context of short-term memory localized in central cortical areas. Several recent studies have shown that early sensory areas also have persistent representations of stimuli which decay over the course of seconds. Traditional mechanisms of short-term memory cannot explain sensory persistence of this form for at least two reasons. (i) Most of those mechanisms are positive feedback models resulting in attractor dynamics, where a transient perturbation results in a quasi-permanent change of system state (attractor dynamics). In contrast, sensory systems respond to a transient by a prolonged return to the original state. (ii) In contrast to higher level areas, dynamics of excitatory connections in early sensory areas tend to be dominated by short-term depression. We develop a theory of short-term persistence in sensory areas by studying negative derivative feedback networks. These networks respond quickly and with high precision to their input. The stability of such networks is known to depend on balancing the strengths of positive and negative feedback. We show that a second condition is required for stability which depends on the relative strengths and time courses of fast (AMPA) and slow (NMDA) currents in the excitatory projections. This condition also determines the response time of the network. We also show that networks which respond quickly to an input are necessarily close to an oscillatory instability which resonates in the delta range and may explain the emergence of absence epilepsy. Persistent activity in early sensory areas operates with two vastly different time courses, showing a fast response during stimulus onset and a slow response after stimulus offset. We show that short-term depression, when acting differentially on positive and negative feedback projections, is capable of dynamically changing the time constant of the underlying network, thus allowing fast onset responses and slow offset responses. We also incorporate this network into grouping models of border ownership selectivity and show that it allows the border ownership signal to persist after the offset of a stimulus and, given appropriate network structure, remap its activity when a visual stimulus shifts its retinotopic position

The Edge of Stability: Response Times and Delta Oscillations in Balanced Networks

<div><p>The standard architecture of neocortex is a network with excitation and inhibition in closely maintained balance. These networks respond fast and with high precision to their inputs and they allow selective amplification of patterned signals. The stability of such networks is known to depend on balancing the strengths of positive and negative feedback. We here show that a second condition is required for stability which depends on the relative strengths and time courses of fast (AMPA) and slow (NMDA) currents in the excitatory projections. This condition also determines the response time of the network. We show that networks which respond quickly to an input are necessarily close to an oscillatory instability which resonates in the delta range. This instability explains the existence of neocortical delta oscillations and the emergence of absence epilepsy. Although cortical delta oscillations are a network-level phenomenon, we show that in non-pathological networks, individual neurons receive sufficient information to keep the network in the fast-response regime without sliding into the instability.</p></div

Response of the balanced network to changes in NMDA/AMPA ratio.

<p>A: Network schematic showing the structure of the rate model used in simulations. Triangular synapses are excitatory and circular synapses are inhibitory. For LIF networks E and I represent populations of 3,200 and 800 neurons respectively with probability of connection between neurons of <i>p</i> = 0.2. B: Simulation of the rate based network for three values of Δ<i>q</i>. At the smallest value, delta oscillations appear (blue line). This value is in the orange range in D, for even smaller values the system is unstable. All rate based networks use <i>k</i> = 1.2, <i>w</i> = 30. C: Same for the LIF network but with <i>k</i> = 0.65 and <i>w</i> = 5.0. D: Rise time in seconds as a function of Δ<i>q</i> for the rate model. Red squares indicate instabilities, the orange segment represents the values of Δ<i>q</i> which generate delta oscillations, and the dashed black line is at the value of Δ<i>q</i> where the rise time is 100 ms. E: Frequency response of the linear system. Delta oscillations start for small negative Δ<i>q</i> (blue) and gamma oscillations (green) appear when Δ<i>q</i> approaches the right instability in D.</p

Dynamics of the balanced network with STD and different usage rates for NMDA and AMPA in the EE projections.

<p>Positive Δ<i>u</i> means higher usage rate for NMDA than AMPA synapses. A: Response of the rate based network with STD to a square pulse imput, beginning at <i>t</i> = 0 and ending at <i>t</i> = 1 s. Parameters are <i>k</i> = 1.2, <i>w</i> = 50, <i>q</i> = 0.5, <i>u</i> = 0.2 and <i>τ</i>_<i>r</i> = 500 ms. B: Rise time of linear networks for the parameters computed from the time dependent synaptic strengths in panel A using the <i>risetime</i> function from Matlab (The MathWorks, Inc., Natick, MA). For Δ<i>u</i> = −0.03, the system is unstable where the blue trace is not shown. C: Temporal trajectories of the network simulated in A as a function of and Δ<i>q</i>. The green circle indicates the starting point of the trajectories while the red circles indicate the end points of the trajectories. The parameters were computed from the STD modulated synaptic strengths at each time point. Shaded areas indicate where the linear system is unstable for the same set of parameters. The AMPA instability corresponds to the left red square in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005121#pcbi.1005121.g002" target="_blank">Fig 2D</a> and the NMDA instability to the right red square. Dashed lines indicate parameters where the linear networks have rise times (RTs) of 0.1 s and 1.0 s. D: Response of the LIF network with STD to a square pulse beginning at <i>t</i> = 0 and ending at <i>t</i> = 1 s. Parameter values are <i>k</i> = 0.65, <i>w</i> = 10, <i>q</i> = 0.5, <i>u</i> = 0.2 and <i>τ</i>_<i>r</i> = 1,000 ms.</p

Dependence of stability, delta oscillations and rise time on NMDA receptors.

<p>A: Values of Δ<i>q</i> for which the network in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005121#pcbi.1005121.g002" target="_blank">Fig 2A</a> is unstable as a function of <i>q</i> and <i>τ</i>^<i>nmda</i>. The dashed black line indicates the maximum absolute value of Δ<i>q</i> possible for each corresponding value of <i>q</i>. The intersections between the dashed black line and colored lines indicate the values of <i>q</i> below which each network is stable for all possible values of Δ<i>q</i>. B: The resonant frequency for each oscillatory instability in panel A. C: The fastest non-oscillatory (critically damped) rise time for each network as a function of <i>q</i> and <i>τ</i>^<i>nmda</i>. The rise time was computed at the value of Δ<i>q</i> where delta oscillations appear or, when <i>q</i> is too small to allow for the emergence of delta oscillations, at the most negative possible value of Δ<i>q</i>. D: The slope of the rise time in the overdamped parameter regime for each 0.01 change in Δ<i>q</i>, see the slope of the line in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005121#pcbi.1005121.g002" target="_blank">Fig 2D</a> where Δ<i>q</i> > 0. The slope was computed from the rise time values for Δ<i>q</i> between 0 and 90% of the value of Δ<i>q</i> at the gamma oscillatory instability. All networks used <i>w</i> = 30 and <i>k</i> = 1.2. Black crosses indicate values of <i>q</i> and <i>τ</i>^<i>nmda</i> for network in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005121#pcbi.1005121.g002" target="_blank">Fig 2B–2E</a>.</p

Frequency responses of the LIF network to constant Poisson input.

<p>Network parameters are as in Figs <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005121#pcbi.1005121.g002" target="_blank">2C</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005121#pcbi.1005121.g004" target="_blank">4D</a>. A: Average over ten runs of the frequency response for each excitatory neuron in the LIF network without STD for different values of Δ<i>q</i>. B: Same as A but with STD. The independent variable is Δ<i>u</i> rather than Δ<i>q</i>. C: Mean of the area under each LIF neuron’s frequency response between 0.5 and 5.5 Hz for the LIF network without STD (left) and between 3.0 and 8.0 Hz for the LIF network with STD (right). The mean value of the integral for Δ<i>q</i> = 0 and Δ<i>u</i> = 0 for each neuron was subtracted for all data points. Error bars are standard deviations.</p

Transient imbalances in the recurrent activity cause the balanced network to act like a damped oscillator.

<p>A: Schematic of a population <i>P</i> receiving excitatory feedback (+), inhibitory feedback (−), and external input <i>I</i>(<i>t</i>). Each recurrent projection has a mix of fast and slow receptors, and projections have equal strength on average. B: Response of the recurrent projections in A to an impulse input, <i>I</i>(<i>t</i>) = <i>δ</i>(<i>t</i>). If the mix of fast and slow currents in the excitatory connection is biased towards the fast receptors relative to the inhibitory connection then excitation is faster than inhibition and the resulting change in <i>R</i> causes a transient increase in input followed by a smaller but longer decrease (blue curve). If inhibition is faster it causes a transient decrease followed by a smaller but longer increase (red). The peak of each response has been normalized to unity. Changes in synaptic strength will scale this response but will not change its shape. We set the fast and slow receptors to be 5 ms and 10 ms in order to allow easy visualization. In all other simulations slow receptors have a time constant of 100 ms unless otherwise noted. C: Schematics indicating the relationship between viscosity for a damped spring and the overall response of the network. D: Response of three systems to a unit step input <i>I</i>(<i>t</i>) = [1 for <i>t</i> > 0; 0 for <i>t</i> ≤ 0], as the effective damping constant increases. Top row: Response of the damped harmonic oscillator derived from panel A. Middle: Response of the network shown in A. Bottom: Response of the full network, see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005121#pcbi.1005121.g002" target="_blank">Fig 2A</a>.</p