New Research In
Physical Sciences
Social Sciences
Featured Portals
Articles by Topic
Biological Sciences
Featured Portals
Articles by Topic
 Agricultural Sciences
 Anthropology
 Applied Biological Sciences
 Biochemistry
 Biophysics and Computational Biology
 Cell Biology
 Developmental Biology
 Ecology
 Environmental Sciences
 Evolution
 Genetics
 Immunology and Inflammation
 Medical Sciences
 Microbiology
 Neuroscience
 Pharmacology
 Physiology
 Plant Biology
 Population Biology
 Psychological and Cognitive Sciences
 Sustainability Science
 Systems Biology
A homeostatic rule for inhibitory synapses promotes temporal sharpening and cortical reorganization

Contributed by James L. McClelland, August 31, 2006
Article Figures & SI
Figures
Data supplements
Moldakarimov et al. 10.1073/pnas.0607589103.
Supporting Information
Files in this Data Supplement:
Supporting Figure 6
Supporting Figure 7
Supporting Figure 8
Supporting Text
Supporting Figure 6Fig. 6. Response time as a function of the strength of the selfexcitatory connection. Red, t_{e} = 3, t_{i} = 5; green, t_{e} = 2, t_{i} = 10; black, t_{e} = 1, t_{i} = 20. b = 1.2 for all curves.
Supporting Figure 7Fig. 7. Phase plane analysis. (A) Nullclines and trajectories for different time scales of the inhibitory unit. Solid lines represent E and I nullclines. The dotted line represents a trajectory with time scale of the inhibitory unit t_{i} = 5; the dotdash line is for t_{i} = 20; and the dashed line is for t_{i} = 80. All connections are fixed at the values α = 3 and β = 1.2. Other parameters fixed and had values as follows: t_{e} = 1, e_{e} = 0.2, e_{i} = 0.15, th_{e} = 1, th_{i} = 0.2, and I_{exte} = 5. (B) Phase plane of the model. Red line, nullcline of the variable E (points where dE/dt = 0); green line, nullcline of the I variable (dI/dt = 0). Point A is a stable fixed point. (ABCDFA is the trajectory of the system stimulated by a short impulse applied to the E cell. (C) Dotted line, the basic E nullcline with α = 3 and β = 1.2; dashed line, the E nullcline with a stronger selfexcitatory connection (α = 3.5 and β = 1.2); solid line, the E nullcline with α = 4 and β = 1. All other parameters are fixed and had values as above. (D) Dotted line, the basic E nullcline with α = 3 and β = 1.2; solid line, E nullcline with stronger selfexcitatory connection (α = 4 and β = 1.2); dashed line, E nullcline for strong selfexcitatory and strong selfinhibitory connections (α = 4 and β = 2.5). All other parameters are fixed and had values as above.
Supporting Figure 8Fig. 8. Response time as a function of the strength of the excitatory connection and the strength of the inhibitory connection. t_{e} = 1, t_{i} = 10, g = 2.5.
Supporting Text
Here, we analyze the response time of the simple preliminary model described in the main text. Our analysis is predicated on the idea that our circuit is set up to produce a transient response to either a transient or sustained stimulus. To achieve this transient response we use a negatively accelerated gain function for excitation like that shown in Eq. 1 and a positively accelerated gain function for inhibition like that shown in Eq. 2. Under these conditions, a transitory sensory response will be produced if the time constant of the excitation,, is smaller than the time constant for the inhibition, . Otherwise, for all but very weak stimuli, the inhibition will simply prevent the excitatory response. Thus, we restrict our analysis to cases where is smaller than. We define the response time of the twounit network as the period during which activity of the E cell is above a fixed threshold. In the simulations reported in the main text, we showed that, if the connections along the selfinhibitory loop are fixed and we change only the strength of the selfexcitatory connection, this strengthening leads to a higher amplitude response but also a longer response time. We begin by considering the generality of this result.
For the simulations of the simplified model in the text, we used and or 20. These values probably exaggerate the differences between excitation and inhibition in the brain. To check that our results are not dependent on this choice, we first consider a case where the two time constants are far more similar: and . As Fig. 6 shows, even in this case, the response time still increases as we increase the strength of the selfexcitatory connection. Indeed, the size of the increase in response time for a fixed increase in α grows larger and larger as α increases, even in this situation.
In what follows, we provide a mathematical analysis of this result based on phaseplane analysis. As already indicated, the analysis assumes that the inhibitory unit is slower than the excitatory unit and that the inhibitory gain function is much steeper than the excitatory one. We also assume that the transient stimulus is very short in duration (a Δ function), so that the locations of the nullclines in the phase plane will remain fixed, with the stimulus setting only initial conditions. For simplicity of the analysis, we also apply the stimulus to the excitatory unit only. Because the inhibitory unit activates much later than the excitatory unit, it is not essential for the analysis whether inputs to the inhibitory unit come from an external input or from the excitatory unit.
The dynamics for E and I cells are described by equations
[A.1]
In Fig. 7A, we colored solid lines to show the nullclines of the system (lines on plane E and I where dE/dt = 0 (red) and dI/dt = 0 (green)). The I nullcline is a monotonically increasing function of E, whereas the E nullcline is a cubic function of E. It is easier to analyze the model for the idealized case in which the I cell is much slower than the E cell (singular perturbation approach). Fig. 7A illustrates advantages of the assumption. There are three trajectories on the phase plane superimposed on the nullclines, for three different choices of the time constant of the I cell. And as we can see as the I cell gets slower, the trajectory moves closer to the E nullcline; in the limit, the trajectory will go along the E nullcline exactly. In this case, it is easy to determine what the amplitude and duration of the response are.
We start at point A in the Fig. 7B (a stable fixed point), and we apply a very short impulse to the E cell. If the impulse is strong enough to move the system beyond point B, the trajectory will continue until it reaches point C on the E nullcline. Because we have assumed that the I cell is much slower than the E cell, the variable I essentially does not change along the line from point A until it reaches point C. From point C the trajectory moves along the E nullcline toward point D. At point D, the trajectory jumps back to the E nullcline (point F), then moves down along the E nullcline toward the fixed point A. Without loss of generality, we can adjust the nullclines such that points A, B, and C will lie on E axis [their coordinates will be (E_{A},0), (E_{B},0), (E_{C},0)]. Now it is easy to determine the amplitudes of responses: the amplitude of the E cell response now is indicated by point C, the point where the E nullcline crosses the E axis. The amplitude of the I cell response is determined by point D, which is a local maximum on the E nullcline.
Based on phaseplane analysis, we conclude that the amplitude of the response of the E cell is determined by the position of the point C along the E axis. The higher point C is along the E axis, the higher the amplitude of the response will be. As we have observed (Fig. 2B), strengthening the selfexcitatory connection alone leads to a stronger response. Now we can explain those observations. We draw nullclines with different strengths of the selfexcitatory connection (Fig. 7C) and observe that point C is shifting to the right as we increase α, meaning more excitation leads to higher amplitude of the E cell response.
Let us now address the question of what determines response time of the model. Because we have assumed that the I cell is much slower than the E cell, it takes much more time for the trajectory to move from point C to point D compared with the time that it takes the system to jump from point A to point C. So the response time is determined mainly by the CD part of the trajectory. As we increase the self excitation by increasing the EE connection (α in the equations), the amplitude of activity of the E unit increases as a response to the external input (Figs. 2B and 7C). But the response time will also increase. On the phase plane, this increase in response time is indicated by the fact that point C has shifted to the right, and point D has shifted up, increasing the distance between C and D (Fig. 7C). In addition, now the CD part of the trajectory comes close to the I nullcline, where the response of the system is quite slow. To get a shorter duration, we need to decrease the distance from C to D and keep the trajectory as far as possible from the I nullcline, because dI/dt = 0 on the nullcline. These considerations do not involve any specific assumptions except the cubic form of the E nullclines. Cubic nullclines can be observed in systems with an fI curve in which frequency increases sublinearly for large currents, which is true for most neurons.
We also draw E nullclines with different selfexcitatory and selfinhibitory connections (Fig. 7D). Here, we see that increasing the excitatory connection with fixed inhibitory strength increases the amplitude of the response and also greatly extends its duration (compare dotted and solid curves). Increasing the inhibitory connection at the higher excitatory strength (dashed curve) does not affect the amplitude. In the regime where the inhibitory time constant is much slower than the excitatory time constant, strengthening the inhibitorytoexcitatory connection has negligible influence on the position of point C but determines the position of point D, moving it down, away from the I nullcline. This transition results both in shortening the CD path and in placing this path farther from the I nullcline. These two effects together result in faster dynamics of the model compared with the case with the fixed inhibition loop.
Response Time Calculation. From Fig. 7D, it is not obvious that the "high amplitude response" is faster than the original "low amplitude response." In this section, we show that, for a given value of the strength of the selfexcitatory loop it is possible to find a value of the strength of the inhibitory connection that makes the "high amplitude response" last for a shorter time than the original "low amplitude response."
What determines the response time of the network? Using notations from figure 7B, it is roughly the time that it takes the system to move from point C to point D. Without loss of generality we have chosen the fixed point A (which is the initial point as well) to be on the E axis. In the limit of very slow I cell dynamics, point C has coordinates (E_{C}, δ), where δ<<1 and we can assume that the coordinates are (E_{C}, 0). This assumption allows us to find E_{C} as the point where the E nullcline crosses the E axis.
Because we apply as an external stimulus a very short impulse, we can write an equation for the E nullcline without the external input term
[A.2]
On the outer branch of the E nullcline E>>1, and therefore F_{e}(x)>>1. Because F_{e} is a monotonically increasing function of its argument, we can conclude that x>>1. Because of the fact that ε_{e}<<1, the excitatory gain function can be approximated as for x>>1.
[A.3]
E_{C} (point C on Figure 7C) can be found as Q(E,I) = 0 for I = 0, which gives us
[A.4]
By solving this equation, we obtain that
. [A.5]
E_{D} (point D on Fig. 7C) can be defined as
and . [A.6]
By inserting E_{D} in Eq. A.3 we can find I_{D} .
. [A.7]
As we already pointed in discussing the simulation of the simple rate model, strengthening the excitatory connection alone increases the amplitude and the duration of the response. Now we can observe the basis for that behavior. First, E_{C} (the amplitude of the response) increases as α (the strength of excitatory connection) increases. Second, the response time of the network is roughly the time that it takes the system to move from point C to point D. By increasing α, we also increase the distance between points C and D.
We have also shown that, by strengthening the inhibitory connection, we decrease the response time of the network. As we have shown, E_{C} and E_{D} depend only on the strength of the excitatory connection, and the strength of the inhibitory connection influences only I_{D}. Strengthening the inhibitory connection moves point D down closer to the E axis, and this transition leads to two effects: first, a path along the E nullcline from point C to point D becomes shorter, and second, point D is now farther away from the I nullcline. This transition also shortens the response time of the network.
The actual time that it takes for the network to move from point C to point D in Fig. 7B can be found by solving the equation
, [A.8]
with an initial point (E_{C},0) and an end point (E_{D}, I_{D}).
Again if we assume that for E>>1,
, [A.9]
where E(I) is the E nullcline
. [A.10]
We approximated the E nullcline between points C and D as a linear function.
, [A.11]
where φ is . [A.12]
By substituting Eqs. A.11 and A.12 into A.9 we obtain
. [A.13]
Putting (for simplicity) θ_{e}=0, θ_{i}=0, and solving this equation, we obtain the time it takes for the system to move from point C to point D.
. [A.14]
Recalling that E_{C}=α, E_{D}=α/2 and I_{D}=α/4β, we find that
and therefore
. [A.15]
T is a function of α/β, so if we increase α alone, the period increases, but if we increase β with α, the period remains constant. Increasing β more will reduce the period. So for given initial α and β, we can easily find values of α and β that can both increase the amplitude of the response and reduce the response time of the system.
Strategy for Connections Adjustments. We have shown so far that the amplitude and the duration of the response can be adjusted by appropriate choices of the strengths of the selfexcitatory and the inhibitory connections. If we want a response to have higher amplitude, we need to increase the selfexcitation, and, in addition, if we want a shorter duration of the response, we have to increase the strength of the inhibitory connection. Given the value of the parameter α Fig. 8 shows how much we need to change parameter β in order to observe temporal sharpening in the simple rate model. All the adjustments we have discussed so far we performed manually. How would a network adjust its connections in order to show temporal sharpening (higher response with shorter duration)? Here we suggest a possible mechanism for such adjustments. If we assume that the network will adjust its connections in such a way that the total activity of the excitatory unit remains constant, then clearly the higheramplitude response must necessarily be shorter than the original lowamplitude response.
Total activity can be estimated as an integral over the area denoted by points A, C, D, and F in Fig. 7B. In a linear approximation, the integral is
. [A.16]
So to keep S constant for higher α, we have to decrease T, which can be done by adjusting parameter β. From Fig.8 we can observe that response time is an increasing function of α and a decreasing function of β. So if we start simulation with weak inhibitory and excitatory connections, and if we increase the excitatory connection, then in order to keep S constant, we have to decrease T below it's initial value, which can be done by strengthening the inhibitory connection. The homeostatic rule for inhibitory connection adjustment that is described in the text ensures that the strengthening of the inhibitory connection is sufficient to produce the sharpening effect.