 # Membrane capacitance Transmembrane potential Resting potential Membrane conductance Constant applied current.

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Membrane capacitance Transmembrane potential Resting potential Membrane conductance Constant applied current

V j increases exponentially from V reset until at which point the cell fires and V j is reset to V reset and the process repeats

Because of this relation between the frequency and I app one can always identify increases in applied current with increases in the frequency or firing rate.

Coupling terms are included as additional current terms on the right hand side of the original equation. One such coupling is chemical synaptic coupling which is modeled by alpha-function current injection. Each time cell k fires, a fixed inhibitory postsynaptic current is injected into cell j. This current has the following form.

The parameter α is the reciprocal of the synaptic time constant and is used to measure the speed of the synaptic dynamics. The parameter q s is a measure of the synaptic strength. The alpha functions are normalized so that the total charge injected with each inhibitory current input equals –q s. During repetitive firing the inhibitory postsynaptic currents sum linearly. An analytical form for the T- periodic total synaptic current injected into cell j can be found by summing the following geometric series when cell k fires T- periodically at times t=nT.

Electrical coupling can be described by two terms. One term is the usual ohmic resistance. Therefore the current flowing from cell k to cell j via electrical coupling is given by: Where g c is the electrical coupling conductance. The second term accounts for the effect of the suprathreshold portion of the spike. Generally LIF models do not include this term but it is important in determining phase-locking patterns. For this model the effect is accounted for by injecting a delta- function current pulse given as follows: β scales the total charge injected by each current pulse.

Note that spikes in a cell generally have a characteristic shape and because membrane conductance is high during a spike, weak to moderate coupling to another cell should have a negligible effect on this characteristic shape of the spikes. Because of this the membrane potential of the cells during a spike will be taken to be: Also a reasonable assumption can be made that during most of the spike.

The total current flowing from cell k to cell j during the spike can be obtained by integrating I spike over the width of the spike (w). Therefore this can be approximated by:

It is useful to then nondimensionalize the model equations to get:

Figure 1. Response patterns of an LIF cell pair coupled by reciprocal inhibition alone: Simulated time courses of transmembrane potential (v) and synaptic currents Isyn are shown for the LIF cell-pair model (Eqs. (5)) with “moderate” inhibitory coupling, gs = 0.2, α = 3. The black and grey curves in the v vs time plots correspond to the membrane potentials of cell 1 and cell 2, v1 and v2, respectively. The black and grey curves in the Isyn vs time plots correspond to the synaptic currents in cell 2 due to the firing of cell 1, Isyn,21, and the synaptic currents in cell 2 due to the firing of cell 1, Isyn,12, respectively. Initial conditions are v1(0) = 0.4, v2(0) = 0.0, Isyn,12(0) = 0.0, Isyn,21(0) = 0.0. (left) When I = 1.1, cells have a relatively low intrinsic frequency and the cells can exhibit stable antisynchronous activity. (right) When I = 1.6, there is a relatively high intrinsic frequency and the system evolves to a synchronous state.

Figure 2. Response patterns of an LIF cell pair coupled by electrical coupling alone: Simulated time courses of transmembrane potential (v) are shown for the LIF cell-pair model (Eqs. (5)) with “moderate” electrical coupling, β = 0.2, gc = 0.2. The black and grey curves correspond to the membrane potentials of cell 1 and cell 2, v1 and v2, respectively. Initial conditions are v1(0) = 0.59, v2(0)= 0.0. (left) When I = 1.1, cells fire at a relatively low intrinsic frequency and the cells can exhibit stable antisynchronous activity. (right) When I = 1.6, there is a relatively high intrinsic frequency and the system evolves to a synchronous state.

Mutual Inhibition Z(t) is the phase-dependent sensitivity function for the LIF model

Figure 3. G-functions (Gs ( φ )) for LIF cell-pair connected by weak inhibition alone ( α = 4.0): Gs ( φ ) determines the phase- locked states of the system in the weak coupling limit (Eq. (8)). φ is the phase difference between the cells. When Gs ( φ ) > 0, φ increases; when Gs ( φ ) 0, then the phase-locked state is unstable. Examples of Gs ( φ ) for three different values of I are shown: (top) I = 1.2; (middle) I= 1.4; (bottom) I = 1.6. Stable synchronous states are indicated by filled diamonds. Stable antisynchronous states are indicated by filled circles, whereas unstable antisynchronous and asynchronous states are indicated by open circles.

Figure 4. Bifurcation diagram for LIF cell-pair weakly coupled with inhibition alone: phase differences of phase-locked states φ ∗ vs applied current I or equivalently φ ∗ vs intrinsic frequency f (see Eq. (2)). α = 4.0. Solid and dashed lines indicate stable and unstable states respectively. I ∗ s indicates the critical value of I at which the antisynchronous state φ ∗ = 0.5 changes stability. For I > I ∗ s, only synchronous activity is stable (white S region), whereas for I < I ∗ s, both synchronous and antisynchronous states are stable (grey AS/S region). The filled diamonds, filled circles and open circles indicate the stable synchronous state, stable antisynchronous state and the unstable asynchronous states respectively for I = 1.2 (Fig. 3 (top)).

Figure 5. Two parameter response diagram for LIF cell-pair coupled with weak inhibition alone, I, α -parameter space: The dashed line plots the location of the critical point I ∗ s in relation to α. (At α = 4, I ∗ s = 1.48 as in Fig. 4). Above the curve, the synapses are fast relative to the intrinsic frequency and both the synchronous and antisynchronous states are stable (AS/S region, grey); Below the curve, the synapses are slow relative to the intrinsic frequency and only the synchronous state is stable (S region, white). As the speed of the synapses ( α ) increases, I ∗ s increases, i.e. the antisynchronous state is stable over a broader range of I for faster synapses.

Electrical Coupling Strongly attracting limit cycle Z(t) is the same as in the previous model of the Mutual Inhibition. β accounts for the effect of the suprathreshold portion of the spike.

Figure 6. G-functions (Gc( φ )) for LIF cell-pair coupled with weak electrical coupling only: For all panels, dashed lines, solid lines and dot-dashed lines correspond to I = 1.05, I = 1.15, and I = 1.3 respectively. Gc( φ ) can be dissected into two parts as in Eq. (10). (top) The portion of Gc( φ ) accounting for the effect of the suprathreshold portion of the spike with β = 0.1. This portion of the G-function always tends to synchronize activity. (middle) The portion of Gc( φ ) accounting for the subthreshold activity (obtained by setting β = 0). This portion of the G-function always tends to desynchronize activity. (bottom) The full G-function shown for β = 0.1 and I = 1.15. The filled diamonds, filled circles and open circles indicate the stable synchronous state, stable antisynchronous state and the unstable asynchronous states respectively.

Figure 7. Bifurcation diagram for LIF cell-pair with weak electrical coupling alone: β = 0.1. Solid and dashed lines indicate stable and unstable phase-locked states respectively. I ∗ c indicates the critical value of I at which the antisynchronous state φ ∗ = 0.5 changes stability. For I > I ∗ c, only synchronous activity is stable (white S region), whereas for I < I ∗ c, both synchronous and antisynchronous states are stable (grey AS/S region). Note that the bifurcation diagram for electrical coupling alone is qualitatively similar to that for inhibition alone (Fig. 4).

Figure 8. Two parameter response diagram for LIF cell-pair connected by weak electrical coupling alone, I, β - parameter space: The dashed curve plots the critical value I ∗ c in relation to β (as in Eq. (11)). For ( I, β ) values below the curve (AS/S region, grey), cells can exhibit either stable synchrony or antisynchrony. For ( I, β ) values above the curve (S region, white), cells can only exhibit stable synchrony. As β increases, I ∗ c decreases. This implies that stronger effects of the suprathreshold portion of the spike promotes synchrony.

Combined Electrical and Inhibitory Coupling

Figure 9. Bifurcation diagram for LIF cell-pair with combined weak electrical coupling and weak inhibition: α = 5.0, β = 0.2 and ρ = 0.5, where ρ is the fraction of electrical coupling, gc/(gc + gs ). Dark solid and dark dashed lines indicate stable and unstable states respectively. I ∗ sc indicates the critical value of I at which the antisynchronous state φ ∗ = 0.5 changes stability. Also, portions of the bifurcation diagrams for electrical coupling alone ( ρ = 1.0, β = 0.2) and inhibitory coupling alone ( ρ = 0.0, α = 5.0) are shown as light curves; I ∗ s and I ∗ c are indicated as well. I ∗ sc must always lie between I ∗ s and I ∗ c, however the ordering of I ∗ s and I ∗ c can be reversed.

Figure 10. Two parameter response diagram for LIF cell-pair with combined electrical and inhibitory coupling, I, ρ -parameter space: This figure summarizes the main results of the paper. I ∗ sc is the critical value of I above which only the synchronous state is stable (S) and below which both the synchronous and antisynchronous states are stable (AS/S). The dashed lines plot the location of I ∗ sc as a function of ρ, the fraction of total coupling due to electrical coupling. I ∗ sc = I ∗ s when ρ = 0 and I ∗ sc = I ∗ c when ρ = 1. I ∗ sc always changes monotonically with ρ. The grey-scale indicates the dominance of the synchronous state, i.e. the probability of the cells evolving to the synchronous state given a random initial phase difference between the cells. The panels depict two qualitatively different situations. (left) “Large” spike effect and “fast” inhibitory synapses, I ∗ s > I ∗ c ( β = 0.3, α = 4.0). In this case, increasing the strength of the weak electrical coupling (increasing ρ ) promotes synchrony. (right) “Small” spike effect and “slow” inhibitory synapses, I ∗ s < I ∗ c ( β = 0.1, α = 1.5). In this case, increasing the strength of the weak electrical coupling (increasing ρ ) promotes antisynchrony.

Figure 11. Two parameter response diagram for LIF cell-pair with combined electrical and inhibitory coupling, ρ, α - parameter space: The dashed lines plot the location of the transition points with spike effect held constant at β = 0.2. (top) I = 1.2, (middle) I = 1.25, (bottom) I = 1.3. For ( ρ, α ) values in grey AS/S region, cells can exhibit either stable synchrony or antisynchrony; for ( ρ, α ) values in white S region, cells can only exhibit stable synchrony. Note that, for I < I ∗ c ( β = 0.2) = 1.28 and a range of ρ, as the speed of the synapse α is decreased, the antisynchronous state can lose stability and then regain stability at lower α (see top and middle panels). This behavior cannot occur for inhibitory coupling alone, ρ = 0.

Though the full system is substantially harder to solve than the weak coupling approximation, one can construct special solutions via analytical methods. When looking for phase-locked states, explicit solution formulae can be obtained between firings and patched together across firings. By doing this the system of differential equations reduces to two algebraic equations for unknown period T and phase difference between 0 and 1. Assumptions Cells fire periodically with period T. Cell 1 fires at times t = nT while cell 2 fires at times t = (n + φ )T where n is an integer. Because of this there are matching conditions that must be met.

Matching conditions on the Membrane Potentials

By using the general solution and the matching conditions, four algebraic equations are obtained.

Figure 12. Results without the weak coupling approximation: Bifurcation diagrams for LIF cell-pair with fixed total coupling strength gtot = gs + gc = 0.1, α = 4.0, β = 0.2 and various ratios of electrical and inhibitory coupling. The overlaid bifurcation diagrams shift systematically with ρ. From right to left, ρ = 0, 0.2, 0.4, 0.6, 0.8, 1. The “tines of the forks” in the bifurcation diagrams corresponding to asynchronous phase-locked states cease to exist at low values of I. This is due to spike-capture synchrony or suppression. The values of I below which the asynchronous states do not exist depend on ρ. This is seen on the tines corresponding to unstable asynchronous-non- antisynchronous states, but it is obscured for antisynchronous state φ = 0.5 due to the overlaying of the bifurcation diagrams.

Figure 13. Results without the weak coupling approximation: Bifurcation diagrams for LIF cell-pair with various total coupling strength and a fixed ratio of electrical and inhibitory coupling ρ = 0.5 ( α = 2.0, β = 0.1). From left to right: the weak coupling limit, gtot = 0.1, 0.2, 0.3 and 0.4). The critical values of I and unstable states for the combined coupling case increase systematically as gtot is increased. As a result of spike-capture synchrony and suppression, the “tines of the forks” in the bifurcation diagrams corresponding to asynchronous phase-locked states cease to exist at low values of I. The values of I below which the asynchronous states do not exist depend on gtot. This dependence is obscured for antisynchronous state φ = 0.5 due to the overlaying of the bifurcation diagrams.

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