1 A coupled excitatory-inhibitory (E-I) pair a neural oscillator

Transcription

1 A coupled excitatory-inhibitory (E-I) pair a neural oscillator A basic element in the basic circuit is an interacting, or coupled, pair of excitatory and inhibitory cells. We will refer to such a pair as an E-I pair. Let x and y be the states or membrane potentials of the excitatory and inhibitory cells respectively, and let the outputs and time constants be g x (x) and g y (y), and, τ x and τ y, respectively. We also use notations α x /τ x and α y /τ y to describe the constants of decay of the neurons to their resting states. According to equation (??, this basic element follow equations of motion ẋ = α x x hg y (y) + I () ẏ = α y y + wg x (x) + I c () where I and I c are external inputs to the two cells respectively, and w > and h > denote the excitatory and inhibitory connection weights. As g x (x) is the output from this pair, the input-output transform of this pair is simply (I, I c ) g x (x). For illustration, consider the example where g x (x = ) = g y (y = ) =, which is usually the case for real neurons when they are at their resting potential. When I = I c =, the static resting states x(t) = y(t) = is an easily verifiable solution. Thus we have one point example (I =, I c = ) g x (x) = in the input-output relationship. The fixed point of this system is ( x, ȳ) where ẋ = ẏ =, satisfies = α x x hg y (ȳ) + I (3) = α y ȳ + wg x ( x) + I c (4) In the example above, x = ȳ = is the fixed point under input (I, I c ) = (, ). With nonzero input I c to the inhibitory cell, when (I, I c ) = (, I c > ), the inhibitory cell raises its state ȳ to ȳ = I c /α y according to equation (4) when g x ( x) =. When the inhibitory cell is charged enough such that g y (ȳ) becomes nonzero, the excitatory cell is inhibited by the inhibitory input hg y (ȳ) to a suppressed state x = hg y (ȳ)/α x according to equation (3). However, output g x ( x) stays zero since g x (.) is an increasing function of its argument. Hence, (I =, I c > ) g x ( x) = is another example in the input-output relationship. The (I =, I c > ) input gives a trivial, open-loop, example to the pair of neurons since, while input I c excites the inhibitory cell, the excitatory cell is passively inhibited by the inhibitory cell without giving nontrivial feedback. The pair becomes a non-trivial close-looped system when the input I to the excitatory cell is large enough to evoke nonzero output g x (x) > despite the the inhibitory input hg x (y) from the cell s partner. Intuitively, the excitatory cell s fixed point x increases with input I and decreases with input I c, i.e., the excitatory cell would be more excited without inhibitory feedback. Meanwhile, the inhibitory cell s fixed point ȳ increases with I c and with I via excitation wg x ( x) from the excitatory cell. In general, the fixed point is where the two curves described by equations (3) and (4) respectively intersect in the x vs. y coordinate system.

2 A: An EI pair C: Fixed point as the interaction output g x (x) I c -, y wg x (x) hg y (y) +, x I y 4 3 ẏ = curve ( x, ȳ) ẋ = curve 3 4 x B: g x (x) and g y (y) g x (x) x g y (y) y D: I g x (x) transform g x (x) h = ց I c = I c = I c = I Figure : The basic circuit element as shown in A: a pair of excitatory and inhibitory cells, interacting with each other, under external inputs (I, I c ), and with output g x (x). B: The piece-wise linear input-output functions g x (x) and g y (y) used as example for illustration. C: Fixed point ( x, ȳ) as the interaction between the two curves. I = 4, I c =, g x (x) and g y (y) as in B. D: The output g x (x) as a function of input I under various input I c to the inhibitory cell, or in the condition without negative feedback h =. Solid curves: I c =, dashed: I c =, dotted: I c =, dot-dashed: h =. The parameters used are x th =, x sat =, g =, y th =, y th =, g =.5, g =, h =, and w = 3.. Gain control in input-output relationship To see how the fixed point ( x, ȳ) depend on input (I, I c ) in more detail, we analyse how small (infinitesimal) changes (δi, δi c ) to input lead to changes (δ x, δȳ) of the fixed point. Differentiating equations (3) and (4) above, = α x δ x hg y (ȳ)δȳ + δi (5) = α y δȳ + wg x ( x)δ x + δi c. (6)

3 The variations of x, y, and g y (y) with input I, under various I c or loop h = conditions. 5 g y (y) 8 y I I 6 x h = 4 I c = ց I c = ց I c = I Figure : Using the same neural parameters as those in fig (), variations of x, y, and g y (y) with I. Solid curve, I c =, dash-dotted: h =, dashed: I c =, and dotted: I c =. Solving the linear equations for (δ x, δȳ), δ x = α y δi hg y(ȳ)δi c α x α y + hwg y(ȳ)g x( x) (7) δȳ = wg x ( x)δi + α xδi c α x α y + hwg y (ȳ)g x ( x) (8) We can understand this solution as follows. First, consider the case when h = or an absent inhibitory feedback to the excitatory cell. Then, δ x = δi/α x, or x = I/α x, the asymptotic state of a cell under input I, as in equation (??). Meanwhile, δȳ = (wg x ( x)δ x + δi c)/α y, or equivalently, ȳ = (wg x ( x) + I c )/α y, the asymptotic state of a cell driven by input wg x ( x) + I c. Second, consider the case w = or when the excitatory cell does not drive the inhibitory one. Then (δ x, δȳ) = (δi c /α y, (δi hg y (ȳ)δȳ)/α x) or equivalently ( x, ȳ) = (I hg y (ȳ))/α x, I c /α y ), the asymptotic states of cells driven by inputs I hg y (ȳ) and I c respectively. When mutual interaction (hw > ), both cells become less sensitive to their direct external input. 3

4 More specifically, sensitivities referred to are δ x δi δȳ δi c δ x δi δȳ δi c = = α x α y when hw = (9) α x+hwg y (ȳ)g x ( x)/αy α y+hwg y (ȳ)g x ( x)/αx when hw () All sensitivities δ x δi, δ x δi c depend on the state values x or ȳ. They are reduced when g x ( x)g y (ȳ) is large, i.e., when the negative feedback loop between the two cells is at a high gain region by high gains, g x( x) and g y(ȳ), of the input-output functions of both cells. Meanwhile, a cell is insensitive to the external input to the other cell if the other cell is at a low gain (g x( x) or g y(ȳ)) region. Most important however is to study how the output g x (x) of this basic circuit element depend on the input, i.e., to obtain the (I, I c ) g x (x) transform, and the sensitivities δg x ( x) δi δi c, δȳ δi, δȳ = g x ( x)δ x δi = δg x ( x) = hg y(ȳ) δi c g x (x) = α y g x ( x) α x + hwg y (ȳ)g x ( x)/α, () y δg x ( x). () δi For concreteness and illustration, consider the following example for g x (x) and g y (y): g y (y) =, if x < x th g(x ) if x th x x sat if x > x sat (3) (4), if y < y th g (y ) if y th y y th g (y ) + g if y > y th The excitatory cell has a piece-wise linear input-output function g x (x), with firing threshold and saturation at x th and x sat respectively, and a gain of g within the operating range x (x th, x sat ). The inhibitory cell also has a piece-wise linear input-output function. This cell has a gain g above the threshold y th, and the gain changes to g at y th > y th which continues into infinity it is just an approximation of a supposed reality where the saturation is beyond the usual operation range, not uncommon for some inhibitory cells. As seen in Fig. (CD), input I drives output g x (x) less effectively under inhibitory feedback h > compared to the open loop condition h =, starting at input values when the inhibitory cell is active g y (y) >. Since the gain g y (y) of the inhibitory cell increases with its state value y, the higher y leads to lower sensitivity δg x (x)/δi, the slope of the g x (x) vs. I curve. The piece-wise linear changes in the slope δg x (x)/δi correspond to the piece-wise changes in g x (x) and g y (y), as a closer examinations of the plots in Fig () indicate that all slope changes are triggered 4

5 by either of the cells passing a threshold or saturation point, x th, x sat, y th, or y th. Note that although sensitivities δ x/δi generally decrease with I, it suddenly increases with I at higher input values to a maximum gain. This happens when x reaches saturation x x sat, such that g x(x) =, effectively severing the transmission to the inhibitory cell the effect of input changes in I and thus disabling the negative feedback in response to the change. This is reflected in the corresponding zero sensitivity δȳ/δi of the inhibitory cell to changes in I. The increased cell state x beyond saturation can not be reflected in the output g x (x) due to a zero gain g x (x) =, see Fig (C). This however, will slow down the response g x (x) to any input changes as the cell takes time to recover from a higher state.. Example: Gain control and contextual modulation of visual inputs The primary visual cortex respond to the contrast of visual input in the way similar to the E-I unit responding to input I. The excitatory cell models one or a local group of pyramidal cells that sends output to higher visual areas. The inhibitory cell models the local interneurons. Visual image input is filtered through retina and other neural circuit before affecting the pyramidal cell in such a way that I relates to the contrast of small local image area, such as a luminance edge, as in Fig. (3). Contributions to I c are from other neighboring neurons or sources outside a cortical area, including top-down feedbacks from higher visual areas and perhaps some components of direct visual inputs. From fig. (D) and equation (), we can see that the contrast gain δg x ( x)/deltai of the E-I unit can be modulated by input level I c which could be controlled by higher visual areas or other neural sources to reflect perhaps contextual or environmental conditions. Given a particular transformation I g x (x), the neuron is most sensitive to a range of contrast I in which the gain δg x ( x)/deltai is large. By adjusting the contrast gain or the transformation I g x (x) within the constraint of the maximum response rate possible g x (x), the neuron could have the optimal dynamic range matched to a particular input environment such as a bright or dim environment, often termed input adaptation level. Response of a visual cortex neuron to an edge within its receptive field has been observed to depend on the visual input context outside and near the receptive field of the neuron. In other words, while the contextual input outside the receptive field does not evoke responses without direct input within the receptive field, it can modulate the neural response to the direct visual input within the receptive field. The contextual modulation can be facilitatory or suppressive, to increase or decrease the neural response respectively. It often depends on the contrast of the direct visual input. For example, neural responses to a weak image edge can often be facilitated by a contextual edge aligned with it. Such a facilitation can serve to detect a faint edge segment, perhaps due to input noise, within a long line or smooth contour. However, the same contextual edge can act to suppress the responses to a strong direct edge segment which is easily detectable. The contextual inputs are direct inputs to other E-I units 5

6 nearby. The neural substrates for the contextual modulation are mainly the neural connections from the excitatory cells from one E-I unit to another. Fig. (3) shows a schematic example of contextual interactions between two aligned edges in an input image. A: mechanisms of influence from contextual visual input A visual cortical unit under study A neighboring unit output g x (x) I c I c -, y -, y +, x +, x Due to contextual input I, change of input from unit (x, y ) to x: I to y: I c Direct input I Contextual input I input image with two contrast edges B: Effect of ( I, I c ) depends on I I/ I c.5 I/ I c = hg y(ȳ)/α y.5.5 Facilitative region Suppressive region 3 4 Contrast input I Figure 3: A: the neural circuit for visual cortical responses to direct I and contextual visual inputs I. Intra-cortical neural connections enable the contextual input I to change the total input to the E-I unit (x, y) from (I, I c ) to (I + I, I c + I c ). B: the effect of the contextual modulation ( I, I c ) depend on the contrast input I. 6

7 The contextual modulation can be modelled as follows. Let (I, I c ) be the input to a particular E-I unit when there is no contextual visual input I. With the contextual input, the total inputs to the excitatory and inhibitory cells in the E-I unit under study become I I + I and I c I c + I c. (5) Consequently, the changes in g x (x) is approximately, according to equation (), g x (x) δg x(x) δi I + δg x(x) δi c I c = δg x(x) ( I hg δi y(ȳ) I c /α y ) (6) It has a facilitative term δgx(x) I caused by I and suppressive term δg x(x) δi δi hg y(ȳ) I c /α y caused by I c. The net result is facilitation or suppression depending on the relative strength of these two terms. It follows that contextual facilitation results when I/ I c > hg y (ȳ)/α y. (7) Otherwise, contextual suppression results. Note that hg y (ȳ)/α y describes the relative sensitivity to input I c compared to that to input I. Accordingly, whether the contextual modulation is facilitatory or suppressive depends on the internal state ȳ. Given a particular contextual modulatory input ( I, I c ), the larger the g y(ȳ), the more likely that the contextual modulation is suppressive. Both physiological and psychophysical data point out that contextual modulation is more likely suppressive when the direct visual input is of higher contrast. This is the so-called contrast dependence of the contextual influence. It can be understood if we assume that higher direct contrast input I gives higher gain g y(ȳ) in the inhibitory interneuron. Since δȳ/δi, this means that the interneuron y normally operates in an operation region of increasing gain, i.e., g y (ȳ) >. The higher gain g y(ȳ) makes the E-I unit more sensitive to suppressive contextual modulation I c than to the facilitatory modulation I..3 A damped neural oscillator One can intuitively see that the E-I pair has a tendency to oscillate. The excited excitatory cell drives inhibitory cell, which increases its response and give inhibitory feedback to the excitatory cell. Subsequently, the suppressed excitatory cell reduces its drive to the interneuron which, in its less excited state, reduces the inhibitory feedback, causing the excitatory cell to rebound. The cycle then starts again and continues until a equilibrium balance is achieved at the fixed point ( x, ȳ). To study oscillations around this fixed point, consider small deviations (δx, δy) from it such that (x, y) = ( x + δx, ȳ + δy). For notation simplicity, we do a coordinate transformation x x x, y y ȳ (8) so that now (x, y) denote the deviation from the fixed point. In this new coordinate system, the fixed point is at the origin. With Taylor approximation g x (x + x) g x ( x) + g x ( x)x g y(y + ȳ) g y (ȳ) + g y (ȳ)y (9) 7

8 equations () and () become ẋ = α x x hg y(ȳ)y α x x hg y (ȳ) + I () ẏ = α y y + wg x ( x)x α yȳ + wg x ( x) + I c () Using equations (3) and (4), we have ẋ = α x x hg y(ȳ)y () ẏ = α y y + wg x ( x)x (3) We perform another temporal derivative d/dt on equation () to have ẍ = α x ẋ hg y (ȳ)ẏ (4) substituting eq. (3) for ẏ = α x ẋ hg y(ȳ)( α y y + wg x( x)x)(5) substituting eq. () for y = (α x + α y )ẋ (6) and we arrive at (hwg x( x)g y(ȳ) + α x α y )x, (7) ẍ + (α x + α y )ẋ + (hwg x ( x)g y (ȳ) + α xα y )x = (8) This is a second order linear differential equation in x, an equation for a damped harmonic oscillator. The second term (α x + α y )ẋ is the friction in the oscillator, arising from the decay or leaky term in the excitatory and inhibitory neurons. The third term (hwg x( x)g y(ȳ) + α x α y )x describes the restoration term in the oscillator. We can compare this equation to that of a harmonic oscillator, with oscillation frequency ω, without damping, ẍ + ω x = (9) which has oscillatory solution x(t) = x()e iωt with initial condition x(). When the decay terms are zero α x = α y =, equation (8) has the same form as a non-damping oscillator, giving an oscillating solution with frequency ω = hwg x ( x)g y (ȳ). To account for non-zero (α x, α y ), assume a solution x(t) = x()e λt, substitute into equation (8) gives [λ + (α x + α y )λ + (hwg x ( x)g y (ȳ) + α xα y )]x()e λt = (3) The λ value is called the eigenvalue of the system, and can be solved as λ = So the solution x(t) becomes (α x + α y )/ ± i hwg x( x)g y(ȳ) (α x α y ) /4 (3) α x=α y= ±i hwg x( x)g y(ȳ) (3) x(t) = x()e αt i ωt where (33) α (α x + α y )/, ω hwg x( x)g y(ȳ) (α x α y ) /4 (34) 8

9 Here, we omitted the solution x(t) e αt+i ωt. This is because in reality, x(t) is a real value, which can be obtained by x(t) = [x(t) + x (t)]/ = x()e αt i ωt + x ()e αt+i ωt = x() e αt cos(ωt + φ) (35) where the superscript denote complex-conjugate of a value, x() and φ are oscillation amplitude and initial phase determined by initial conditions (x(), ẋ()), or equivalently (x(), y()), from which we can deduce ẋ() via equation (3). For analytical convenience, we will use the convention of complex values e iωt to denote oscillations unless stated otherwise. Given solution x(), equation () gives y(t) = ( α x x ẋ)/[hg y (ȳ)] (36) = (α y α x )/ + iω hg x(ȳ) x(t) y()e αt iωt (37) (αy α x ) hence y() = x() / 4 + ω hg x(ȳ) e iθ (38) where θ = arctan(ω/(α y α x )) (39) which means that y(t) has the same damped oscillation as x(t), but with a oscillation phase lag of θ and an oscillation amplitude scaled by a factor (αy α x) / 4+ω hg x (ȳ). When α = α x = α y, θ = π/, so y(t) is phase lagged by a quarter oscillation cycle, and with amplitude scaled by a factor hwg x( x)g y(ȳ). For simplicity in our discussion, we use mainly the simple case when α = α x = α y, we have ω = hwg x( x)g y(ȳ). Fig. (4A) shows an example of x(t) and y(t) pair, where the phase lag was apparent. The quarter-cycle phase difference also means that trajectory (x(t), y(t)) traces out a shrinking spiral in x-y phase space and converge to the fixed point ( x, ȳ), see Fig. (4B). If the damping term is zero α =, this spiral becomes an elipse, and the oscillation amplitude will be sustained..3. An alternative approach to the solution The steps from equation () to equation (39) to obtained solutions [x(t), y(t)] for the linear equations () and (3), can be replaced by a standard approach (which we will often use) in solving a set of linear differential equations. Treat (x, y) as components of a two dimensional vector, we have ( ( ) ( x α hg d/dt = y (ȳ) x y) wg x( x) (4) α y) In general, linear differential equations ż = Mz, where z is a N dimensional vector and M is a N N matrix with constant elements, have the solution z = sum N i= c iv i e λit. Here, v i is the i th eigenvector of the matrix M, λ i is the corresponding eigenvalue such that Mv i = λ i v i, and c i are 9

10 A: x(t) and y(t) dynamics.6.4. x(t) y(t) t B: the trajectory x vs.y for A y ( x, ȳ) x Figure 4: The oscillation dynamics of an EI pair. h =, w = 4, α x = α y =., I = 6, I c =, both g x and g y take the form g(u) = /(+e 4(u.5) ). A plots x(t) and y(t) starting from initial condition x() = y() =, note the phase difference between x(t) and y(t). B plots this spiraling trajectory in the x -y coordinate. Note that here we plotted x, y in the coordinate system before the transformation (x, y) (x x, y ȳ). constants such that sum N i= c iv i = z(t = ) is the initial condition. Applying to our E-I pair, the eigenvalue and eigenvectors of the matrix ( ) α hg M = y (ȳ) wg x ( x) α (4) are solutions of equation [λ + (α x + α y )λ + (hwg x( x)g y(ȳ) + α x α y )] = (4) which is just like equation (3), with its two solutions expressed in equation (3), one for each of the eigenvectors. The eigenvector for one the eigenvalue λ = α x i hwg x( x)g y(ȳ) is ( ( ) x y) i (wg x ( x)/(hg y (ȳ)) (43) which is the same as in equation (39), expressed as a phase and amplitude relationship between x and y. The other eigenvalue and eigenvector are simply complex conjugates of this one. So we arrive at the same solutions..4 A Lyapunov function for the E-I pair When the oscillation amplitude is too large, the linear approximation is not accurate, and thus the sinusoidal oscillation (in condition α = ) becomes non-sinusoidal. However, it can be show that there is a Lyapunov function Using the original coordinate (x, y) rather than deviations from the fixed

11 A: Three [x(t), y(t)] traces x(t): solid, y(t): dashed.8 g x (x) t B: Their trajectories in x-y space each traces a contour of a constant L(x, y) value. x.6 y.6 y L = L =.96 ւ L = g y (y) x Figure 5: The non-damped oscillations around the fixed point in Fig. (4) when the decay terms were turned off, i.e., α =. The external inputs (I, I c ) = (4.7,.) are re-adjusted to maintain the same fixed point under α = condition. All other parameters are the same as in Fig. (4). Three different initial conditions (x(), y()) gives three oscillation traces, plotted as temporal traces in A (note three different scales used on the y-axis on the three plots), and in x-y phase space in B. The Lyapunov functional values are L(x, y) = ,.96,.94 for larger and larger oscillations. In A, the oscillation traces are less sinusoidal for larger amplitude oscillations. This is manifested in less elipse like trajectories in B. Also plotted in the upper right and lower left are the g x (x) and g y (y) function curves aligned to the plot in B to indicate the operation range of the various motion trajectories relative the the linear and non-linear regions of the gain functions. points, L(x, y) = w x x du[g x (u) g x ( x)] + h y ȳ du[g y (u) g y (ȳ)] (44) Note that since u x or u ȳ is always of the same sign as [g x (u) g x ( x)] or [g y (u) g y (ȳ), respectively. Each integral in L(x, y) is thus non-negative, increases with x x and y ȳ respectively, and becomes zero only when

12 x = x or y = ȳ respectively. L(x, y) evolves in time as dl(x, y)/dt = ẋdl/dx + ẏdl/dy = w[g x (x) g x ( x)][ α x (x x) h(g y (y) g y (ȳ))] +h[g y (y) g y (ȳ)][ α y (y ȳ) + w(g x (x) g x ( x))] = α{w(x x)[g x (x) g x ( x)] +h(y ȳ)[g y (y) g y (ȳ)]} (45) The last inequality is arrived by noting that x x and y ȳ are always of the same sign as [g x (x) g x ( x)] and [g y (y) g y (ȳ)] respectively. When α >, dl/dt <, and the decreasing L(x, y) shrinks down the magnitude x x and y ȳ, which conceptually represent the oscillating amplitude around the fixed point ( x, ȳ). When α =, dl/dt = even when (x x, y ȳ) while (ẋ, ẏ). This means that the motion trajectory is a closed curve surrounding ( x, ȳ) in (x, y) space, and L(x, y) is a constant on the closed curve. The dynamics is then a perpetual cycle. The closed curve is an elipse and the oscillation cycle is sinusoidal when both g x (.) and g y (.) are linear functions. In Fig. (5), we can see the positions of the motion trajectories relative to the linear and nonlinear regions of the g x (x) and g y (y) function. As the trajectory spans a more or less linear region, the trajectory is more or less elipsoidal. The two terms in L(x, y) above can be intuitively viewed as the kinetic and potential energies in the oscillation. The energies decay with friction in the system. To view this more clearly, take the case when both g x (x) = a(x x ) and g y (y) = b(y y ) are linear, with constants a, b, x and y. Then, L(x, y) = wa (x x) + hb (y ȳ) (46) which resembles the familiar kinetic and potential energy terms in harmonic oscillators when x x and y ȳ are viewed as displacement and velocity respectively of the oscillator. In the case when there is no damping, ẋ = hb(y ȳ), then the form L(x, y) = wa (x x) + hbẋ (47) becomes more obviously familiar to summation of kinetic and potential energy of an oscillator, whose frequency is hwab = hwg x ( x)g y (ȳ)..5 Oscillation frequency It is apparent from ω = hwg x( x)g y(ȳ) (48) that the oscillation frequency ω can be tuned by changing the synaptic connection strength h and w, or by changing ( x, ȳ) via external input (I, I c ). To have larger oscillation frequencies, it helps to place ( x, ȳ) at the locations of higher gain (g x ( x), g x (ȳ) in the respective functions. This is demonstrated in Fig. (6), where three different input conditions give three different ( x, ȳ)

13 .5 4 A: motion traces x(t): solid; y(t): dashed three y(t) (red) and x(t) (blue) traces for different input I ( x, ȳ) = (.5,.5) ( x, ȳ) = (, ) ( x, ȳ) = (.5,.5) t B: the trajectories x vs.y for traces in A y Increasing input I x 4 6 Figure 6: Input control of the oscillation frequencies by shifting the fixed point ( x, ȳ). h = 3, w = 8, α =. Both g x and g y take the form g(u) = /( + e 4(u.5) ). Three different fixed points: ( x, ȳ) = (.5,.5), (, ), and (.5,.5) are created by three different input (I, I c ) = (.4,.356), (4.58,.464), and (6.5,.5) respectively. All three conditions start with the same initial condition (x(), y()) = (, ). Note that the oscillations for the second condition is less sinusoidal like as the third condition, which has its fixed point at a more linear region of the g x and g y. B plots the three spiraling trajectories (dot-dashed, dashed, and solid curves for the three ( x, ȳ) conditions respectively) in the x -y coordinate for the three conditions. Note the increase in oscillation frequency as the fixed points are increased in value to reach higher gains g x( x)g y(ȳ). and thus three differnt oscillation frequencies. In all conditions, the same initial state value (x(), y()) is below the respective fixed point. Within the same time duration, each motion trajectory approaches the respective fixed point, with the more spirals, i.e., faster oscillations or larger frequencies, around the largest fixed point. Note that when the oscillation frequency is too small, as in the example when ( x, ȳ) = (.5,.5), oscillation becomes unapparent while the amplitude decays too fast compared with that of the oscillation. Specifically, let the deviation from ( x, ȳ) be approximately, (x, y) e αt iωt. The amplitude decays exponentially with a time constant τ = /alpha, while the oscillation has a period T = π/ω. When τ T, oscillation is negligible. The membrane time constant of a neuron is /α, which in many neurons are of the order of milliseconds. Since our unit time used in the figures is this time constant, the periods of oscillations in Figs (4), (5), and (6) are also on the order of milliseconds, which means an oscillation frequency of Hz. This frequency is comparable to or higher than most neural oscillations seen in the brain, e.g., gamma and theta oscillations are on the order of 4 Hz and Hz respectively, or 5 and milliseconds of oscillation period. We used small enough oscillation period T in our figures in order to demonstrate the oscillation behavior of our simple neural oscillator clearly. Fig. (7) shows different oscillation frequencies by using different synaptic weights h and w while 3

14 keeping ( x, ȳ) fixed. A: higher frequency oscillations x(t): solid; y(t): dashed B: lower frequency oscillations x(t): solid; y(t): dashed t t Figure 7: Higher (A) and lower (B) oscillation frequencies are obtained by tuning the synaptic connection weights h and w. h =, w = 4 in A while h = w = 4 in B. α =. Both g x and g y take the form g(u) = /(+e 4(u.5) ). The fixed point ( x, ȳ) = (.3,.) in both plots, achieved by having I = 5.9 in A and I =.3 in B, while I c in both A and B. Both plots start with the same (x(), y()) = (, ). Note that the oscillation traces are less obvious when the oscillation period is comparable or longer than the time constant for the oscillation amplitude to decay. Of course, when the oscillation amplitude is too large, the trajectory covers the phase space where the gains g x ( x)g y (ȳ) are not constant. This also makes the oscillation frequency amplitude department, even when the fixed point ( x, ȳ) is unchanged. This is apparent in Fig. (6) within a single oscillation trace. For instance, in the oscillation trace where ( x, ȳ) = (.5,.5), the oscillation period is larger for larger amplitude oscillations, so the oscillation frequency increases while the amplitude decays to zero. This is because the fixed point ( x, ȳ) = (.5,.5) is actually the location where both g x ( x) and g y (ȳ) are maximum, and that either gain decreases with increasing distance x x or y ȳ respectively. Hence, larger amplitude oscillations leads to smaller g x( x)g y(ȳ) and the oscillation frequency..6 Response to non-stationary or oscillatory inputs resonance Consider an E-I pair in the situation when I c is static while I is a function of time I(t). It can always be be composed of a static component and a summation of sinusoidal components I(t) = Ī + ω I(ω)e iωt (49) where Ī does not change with time, I(ω) represent the amplitude and phase of the sinusoidal component with frequency ω. Let ( x, ȳ) be the 4

15 fixed point under static input (Ī, I c). Following procedures from equation (8) to equation (3), one see that small deviations (x, y) from ( x, ȳ) approximately follow equations ẋ = αx hg y (ȳ)y + ω I(ω)e iωt (5) ẏ = αy + wg x( x)x (5) Also, analogous to equation (8), we have equation ẍ + α x ẋ + (hwg x( x)g y(ȳ) + α )x = ω (α iω)i(ω)e iωt (5) For simplicity, we consider a single sinusoidal input component such that I(t) = Ī + Ie iωt, where for simplicity of notation I now represent the amplitude and phase of the sinusoidal drive. Intuitively, the external drive would make the neuron respond with the driving frequency. Substitute x(t) = xe iωt (53) into equation (5) under single sinusoidal drive, we have x = (α iω)i/[ ω iωα + (hwg x ( x)g y (ȳ) + α )] α iω = I ωo (ω + iα) (54) where ω o = hwg x ( x)g y (ȳ) is the intrinsic frequency of the neural oscillator s intrinsic dynamics e αt iωot without the external inputs. This means, ignoring the initial transients, the response follows the external oscillatory drive in the same frequency, with an oscillation amplitude α x = I + ω (ω + α ωo) + 4ωoα (55) and a phase lead relative to the drive as tan (ω/α) tan [(ωα)/(ωo + α ω )] if ωo φ = + α ω > tan (ω/α) π + tan [(ωα)/ ωo + α ω (56) ] otherwise We define α Q(ω) + ω (ω + α ωo) + 4ωoα (57) as the sensitivity to the external drive. Then x = IQ(ω)e iφ (58) Note that the sensitivity depends on ω. Hence, the response is tuned to the driving frequency. The optimal frequency ˆω to which the neural oscillator is the most sensitive is easily seen from equation (57) as satisfying ˆω ω o α (59) 5

16 which minimizes the denominator in Q(ω). So for small damping such that α ω o, the optimal driving frequency is close to the intrinsic frequency frequency ω o. The tuning width of Q(ω) is roughly, from the denominator again in equation (57), ω ω o α, which means, ω ω o α/ω. (6) Hence, tuning width increases with increasing damping α. In other words, small damping enables a neural oscillator to be very selective to the driving frequency. A high selectivity leads to a resonance phenomena, meaning that the response is insignificant unless the driving frequency is close to the optimal frequency ω ˆω. From equation (56), we see that the driven oscillator x slightly phase lead the driving force for small ω but lags the driving force for larger ω. Near resonance when ω = ω α, x and I are exactly in phase. It is not intuitive to imagine that the driven oscillator x(t) should phase lead the input I(t). In a physical situation when an oscillator is driven by an external oscillatory force in the sense that the force contributes to the acceleration of the oscillator as in Newton s law, the driven oscillator normally phase lags the driving force. This physical situation is described by ẍ + αẋ + (ω o + α )x = F(t) Fe iωt (6) where F is the amplitude and phase of the driving force. Comparing this equation with equation (5), we see that F corresponds to (α iω)i(ω). We can then arrive at x = FQ(ω)e iφ with { tan φ = [(ωα)/(ωo + α ω )] if ωo + α ω > π + tan [(ωα)/ ωo + α ω (6) ] otherwise Hence, the oscillator x(t) always phase lags the driving force F(t). The sensitivity is also slightly changed, Q(ω) (ω + α ωo) + 4ωoα (63) which still displays the resonance or frequency tuning phenomena with roughly the same oscillation frequency and tuning width. Note that when the oscillatory drive is from the input I c such that I c (t) = Īc + I c e iωt, then F(t) = I c e iωt. Hence, in general, let I(ω)e iωt and I c (ω)e iωt (64) ω> be the AC components of the input to the excitatory and inhibitory cells respectively, the the AC component of the drive to the oscillator is then F(t) = F(ω)e iωt = dω[(α iω)i(ω) + I c (ω)]e iωt. (65) ω> ω> Oscillatory inputs to neural oscillators can be seen in the example of olfactory cortex, which is intrinsically but not spontaneous oscillatory, and which has oscillatory inputs from the olfactory bulb. Both the intrinsic oscillation frequency of the olfactory cortex and the inputs from the bulb are of the gamma frequency range. ω> 6

17 A: Input and response traces x(t): solid; I(t): dashed ω = ω = 6.5 ω = 5 5 t B: the frequency tuning Q(ω) Q(ω) C: the cycle lead φ/(π) φ/(π) ω ω Figure 8: The oscillator as in Fig. (4) is driven additionally by oscillating input Ie iωt for I =. A: Ie iωt (scaled down) and x(t) for three different ω =, 6.5,. While the input amplitude is the same, output response amplitude changes with ω. The resonance frequency is close to ω = 6.5. Note the initial transient in the intrinsic frequency of the neural oscillator before the asymptotic oscillations in the frequency of the input. B: The Q(ω) function as in equation (57) with ω o = 6.5 and α = (this α is used in A). C: The corresponding cycle lead φ(ω)/(π) function with the same ω o and α. A self-excitatory neural oscillator The damped oscillator in the last section can become sustained if the excitatory cell is self-excitatory, as ẋ = α x x + Jg x (x) hg y (y) + I (66) ẏ = α y y + wg x (x) + I c (67) 7

18 This is the same as equations () and () except that an additional term Jg x (x) is added in the equation for ẋ. Following the same procedure as before, the deviation of x from the fixed point ( x) can be shown to follow the equation, in the linear approximation: ẍ + (α x + α y Jg x( x))ẋ + (hwg x( x)g y(ȳ) + (α x Jg x( x))α y )x = (68) This is the same as equation (8) except that α x in equation (8) is now α x Jg x( x). Intuitively, this means the damping of this oscillator stemming from α x is reduced by the amount Jg x ( x), as can also be seen in equation (66). Hence, the effects of reduced damping and self-excitation could be qualitatively be equivalent. However, if J is sufficiently large, the effective damping α x Jg x ( ) can be negative, and when this is sufficiently negative, the oscillator can have sustained rather than damped oscillation. The linear solution is x(t) = x()e λt, and as before, λ is the eigenvalue of the system satisfying equation (compare with equation (3)): λ + (α x + α y Jg x( x))λ + (hwg x( x)g y(ȳ) + (α x Jg x( x))α y )] = (69) Hence λ = (α x + α y Jg x ( x))/ (7) ±i hwg x ( x)g y (ȳ) (α x Jg x ( x) α y) /4 (7) α x=α y=α α + Jg x ( x)/ (7) ±i hwg x ( x)g y (ȳ) (Jg x ( x)) /4 (73) x(t) = x()e (α Jg x ( x)/)t i ωt (74) where ω hwg x( x)g y(ȳ) (Jg x( x)) /4 (75) This oscillator can sustain its oscillation when there is sufficient self excitation such that Jg x ( x)/ > α. At the same time, the oscillation frequency is also reduced from the value ω hwg x( x)g y(ȳ), whether or not the self-excitation is sufficient to make the oscillation self-sustaining. The value of the negative damping, Jg x ( x) depends on the fixed point x, which is controlled by the external input (I, I c ). We have seen in section (.5) that oscillation frequency can be modulated by inputs. Here, whether the oscillator can have sustained oscillation can also be modulated by external input. For example, given a fixed I c to the inhibitory cell, increasing the input I to the excitatory raises x. Hence, when I is so weak that x is below threshold and g x ( x) =, the slope or gain g x( x) is zero and the oscillator is damped. As I increases, x passes the threshold to achieve a larger gain g x ( x), spontaneous oscillation is possible when this gain is sufficient. In Fig. (9)A, three different traces of x(t) were plotted for input values (I, I c ), which correspond to increasing levels for the fixed point ( x, ȳ). Sustained oscillation is achieved for the intermediate fixed point value ( x, ȳ) which correspond to the highest gain g x( x) possible. When the fixed 8

19 A: motion trajectories x(t) around three different mean x x(t): solid; x(t): dashed ( x, ȳ) = (.85,.85) ( x, ȳ) = (.5,.5) ( x, ȳ) = (.5,.5) B: The g x (x) or g y (y) function.5 g x(x) or g y (y) 3 4 t x or y C: x(t) (top) and I(t) (bottom) D: An excitatory-inhibitory pair x(t) I(t) t I c -, y wg x (x) hg y (y) output g x (x) +, x J Figure 9: Input control of the emergence of neural oscillation by shifting the fixed point ( x, ȳ). J = 3, h = 4, w = 4, α =. Both g x and g y take the form g(u) = /( + e 4(u.5) ). The dynamics follows equations (66) and (67). A: Three different fixed points: ( x, ȳ) = (.,.5), (.5,.5), and (.85,.85) are created by three different input (I, I c ) = (.35,.36), (,.5), (.7,.36) respectively. All three conditions start with the same initial condition (x(), y()) = (, ). Note the sustained oscillation for the second condition. B: The g(u) function and the location of the three fixed points x corresponding to the motion traces in A. C: the trajectory x(t) when I(t) slowly changes in time. Initial condition (x(), y()) = (, ). I(t) =. + [ cos(πt/t)]+noise(t), where T = 5, and noise(t) is random number uniformly sampled from range (,.), changes with time only once every time interval of length./α. I c =.5. Note from the plot of I(t) that noise amplitude is very small (and almost imperceptible with the plot resolution) over the smooth sinusoidal variation of I(t). The dotted curve in the upper plot traces out (the approximate) x, which evolves in the same time scale as I(t). I 9

20 point x is too high, it reaches towards the saturating region for the function g x (x), and thus the gain g x( x) is lowered, making the oscillator damped again. If we again examine the quantity for the Lyapunov function as defined in equation (44): L(x, y) = w x It now evolves in time as x du[g x (u) g x ( x)] + h dl(x, y)/dt = ẋdl/dx + ẏdl/dy y ȳ du[g y (u) g y (ȳ)] (76) = w[g x (x) g x ( x)][ α x (x x) h(g y (y) g y (ȳ))] = +w[g x (x) g x ( x)][j(g x (x) g x ( x))] +h[g y (y) g y (ȳ)][ α y (y ȳ) + w(g x (x) g x ( x))] = α{w(x x)[g x (x) g x ( x)] +h(y ȳ)[g y (y) g y (ȳ)]} +wj[g x (x) g x ( x)] (77) We can see that the first term α{...} is non-positive since (x x)[g x (x) g x ( x) and h(y ȳ)[g y (y) g y (ȳ)] for all (x, y), but the second term wj[g x (x) g x ( x)] is always non-negative. Hence, L(x, y) is not guaranteed to be always decreasing in the dynamics. This means it is no longer a Lyapunov function when J is sufficiently large. When (x, y) is sufficiently close to ( x, ȳ), one can approximate g x (x) g x ( x) g x ( x) x, and g y (y) g y (ȳ) g y(ȳ) y, where ( x, y) (x x, y ȳ). Then, dl(x, y)/dt w(jg x( x) α)g x( x) x αhg y(ȳ) y (78) when Jg x ( x) > α, dl(x, y)/dt can be positive for some values of ( x, y), though the value of L(x, y) over the motion trajectory could still rise and fall. The self-excitation in the excitatory cell has been used in many models of neural oscillators in order to produce the neural oscillations observed in physiological experiments, as this is the easiest way to generate oscillations without using network effects (between different neural oscillators) or using cellular level mechanisms (not modelled here) which could also produce oscillatory tendency.