# Effects of Synaptic Plasticity on Phase and Period Locking in a Network of Two Oscillatory Neurons

- Zeynep Akcay
^{1}, - Amitabha Bose
^{1}and - Farzan Nadim
^{1, 2}Email author

**4**:8

https://doi.org/10.1186/2190-8567-4-8

© Z. Akcay et al.; licensee Springer 2014

**Received: **20 September 2013

**Accepted: **25 February 2014

**Published: **29 April 2014

## Abstract

We study the effects of synaptic plasticity on the determination of firing period and relative phases in a network of two oscillatory neurons coupled with reciprocal inhibition. We combine the phase response curves of the neurons with the short-term synaptic plasticity properties of the synapses to define Poincaré maps for the activity of an oscillatory network. Fixed points of these maps correspond to the phase-locked modes of the network. These maps allow us to analyze the dependence of the resulting network activity on the properties of network components. Using a combination of analysis and simulations, we show how various parameters of the model affect the existence and stability of phase-locked solutions. We find conditions on the synaptic plasticity profiles and the phase response curves of the neurons for the network to be able to maintain a constant firing period, while varying the phase of locking between the neurons or vice versa. A generalization to cobwebbing for two-dimensional maps is also discussed.

## Keywords

## 1 Introduction

The output of a neuronal network, determined in part by the relative spiking times of its individual neurons, depends on the coordinated activity of its neurons. Observed phase relationships result from the combined effects of individual cells and synaptic connections whose properties change dynamically. For example, individual neurons in a network can differ in their intrinsic properties, being silent, spiking or bursting; different neurons can have different responses to the synaptic inputs they receive, and the synaptic inputs themselves can differ widely. These different characteristics all play a role in determining the resulting network activity. Determining how these dynamically varying components work together to influence the network activity is a question of considerable interest.

Many studies have explored the question of how period and phase are determined in an oscillatory neuronal network [1–7]. One of the main tools used in these studies is the phase response (or resetting) curve (PRC) of an individual neuron. The PRC measures how the phase of firing of an oscillatory neuron changes as a function of perturbations that it receives at different phases of its oscillation. In prior work [8], the PRC has been used to define a 1D map that measures the degree of network synchrony. This map allows for the analysis of the network activity in a reduced system by considering only the effect of the synaptic inputs on cycle length, rather than considering multiple dynamic variables. Several studies used similar methods to study the activity of neuronal networks [2, 8–12]. PRC-based maps were also used to incorporate some properties of neurons or synapses. This approach was applied to understand synchronization of adapting neurons [2, 5] as well as the effect of conduction delays on network synchrony [1, 13].

In the current study, we are interested in predicting phase-locking by deriving maps that combine PRCs with information arising directly from synapses that display frequency-dependent short-term plasticity. Increases in presynaptic firing frequency can strengthen (facilitation) or weaken (depression) a synapse [14]. Some synapses show a combination of both, in which case the maximal synaptic amplitude is achieved at a specific presynaptic frequency [15] referred to as the preferred frequency of the synapse. Synaptic plasticity can be described with models having two variables, one for depression and the other for facilitation [15, 16].

The main advance in our work is the derivation of tools for analyzing higher-dimensional maps that incorporate the effects of synaptic plasticity and provide predictions on circumstances under which an oscillatory network of neurons will phase-lock and at what period. In particular, we consider a network of two neurons, mutually coupled by inhibition in which the synaptic strength is frequency dependent. In deriving these maps, we must not only track the phases of each cell, but also the strength of each synapse. As a result, the 1D map that sufficed in prior studies needs to be replaced with 2D or 3D maps. For 2D maps, we derive a geometric method that generalizes the idea of cobwebbing. Namely, we show how iterations of the map can be tracked through different 2D surfaces. Moreover, projections of these surfaces onto a common plane yields two curves whose intersection is a fixed point of the map that corresponds to a phase-locked solution. We derive conditions on the PRCs and the parameters that govern synaptic plasticity of the neurons to show how a network can have a range of parameters over which the network period remains constant, but the phase of locking between cells changes, or vice versa. We also show that the methods derived apply to networks that are heterogeneous either in the intrinsic properties of individual cells, in their synapses, or both.

## 2 Model and Methods

### 2.1 Dynamics of Neurons

*L*), potassium (

*K*) and calcium (

*Ca*) current. The

*K*current is driven by a dynamic activation variable

*w*, while the

*Ca*current depends on an instantaneous function ${m}_{\mathrm{\infty}}$ of the membrane voltage (

*V*). When the neuron is synaptically coupled to another neuron, the synaptic current received from the other cell is also included in the equation governing the membrane voltage. For two M–L neurons coupled with synaptic inhibition, the equations for voltage

*V*and

*K*activation variable

*w*are given by

for $i,j=1,2$, $i\ne j$, where ${m}_{\mathrm{\infty}}(V)=0.5(1+tanh((V-{V}_{a})/{V}_{b}))$, ${w}_{\mathrm{\infty}}(V)=0.5(1+tanh((V-{V}_{c})/{V}_{d}))$ and ${\tau}_{w}(V)=1/(\varphi cosh((V-{V}_{c})/2{V}_{d}))$. The conductances (in nS) are ${\overline{g}}_{L}=2$, ${\overline{g}}_{K}=8$, ${\overline{g}}_{\mathit{Ca}}=4$, the reversal potentials (in mV) are ${E}_{L}=-60$, ${E}_{K}=-84$, ${E}_{\mathit{Ca}}=120$ for the leak, potassium and calcium currents, respectively. The synaptic reversal potential ${E}_{\mathrm{syn}}$ is −80 mV, modeling an inhibitory synapse. Due to the presence of the Heaviside function $\mathrm{H}(V-{V}_{\mathrm{th}})$, the synapses are all-or-none and activate (deactivate) instantaneously when the presynaptic voltage is above (below) the synaptic threshold ${V}_{\mathrm{th}}=0\text{mV}$. The results described in this study remain qualitatively similar if the value of ${V}_{\mathrm{th}}$ is changed.

Below we will provide more details of the synaptic conductance ${g}_{\mathrm{pre}\to \mathrm{post}}$. We change the applied current ${I}_{\mathrm{app}}$ (in pA) between 41.2 and 44.9 to obtain a set of intrinsic periods (in ms) ranging between 100.3 and 180.83. The rest of the model parameters are $C=20$ pF, $\varphi =0.067$ (dimensionless), and ${V}_{a}=-1.2$, ${V}_{b}=18$, ${V}_{c}=12$, ${V}_{d}=17.4$ in mV. Throughout the paper, units for time are in msec.

### 2.2 Phase Response Curves

*dt*after the firing of the cell. This yields a phase $\varphi =dt/{P}_{0}$ of the perturbation. Denote by $\tilde{P}$ the time between when a cell fires prior to a perturbation and the subsequent firing of the cell when a perturbation is given at phase

*ϕ*. Then we define the PRC as

The reference point to compute the PRC is chosen to be when *V* crosses ${V}_{\mathrm{th}}$ in the positive direction. Note again that this method of computing the PRC is different from computing the iPRC of a spiking neuron which yields a strictly Type 1 PRC. The PRC we obtain is very similar, but there is a region of the PRC that is positive near small stimulus phases due to the longer active duration of the neuron.

#### 2.2.1 Selection of PRCs

In order for our analytical estimates to match the results of numerical simulations of the model, we took advantage of the computability of a PRC for the M–L neuron. In each iteration, we numerically computed the response of a neuron to a synaptic input of a specific strength at a specific phase. Although this method yields accurate results, it is computationally slow and it is almost impossible to implement on biological neurons. For this purpose, we created a meshed PRC measured at discrete phase points and for a discrete set of predetermined synaptic strengths. We used mesh sizes of 0.1 for the phase and 0.0125 for the synaptic strength to obtain a total of 77 points of numerically computed phase response values. The responses to the phases and strengths not on the mesh points were calculated by linear interpolation.

### 2.3 Model for Synaptic Plasticity

*r*) and a facilitation variable (

*u*). The depression variable

*r*represents the amount of available synaptic resources, while the variable

*u*represents the amount of utilized synaptic resources. They change according to the activity of the presynaptic cell and together determine the synaptic strength. These variables obey the following dynamics:

When the membrane voltage of the presynaptic cell is above the synaptic threshold ${V}_{\mathrm{th}}$, the depression variable *r* approaches 0 with time constant ${\tau}_{1}$, representing the depletion of available synaptic resources. During this time interval, the facilitation variable *u* approaches 1 with time constant ${\tau}_{3}$ representing the increase in utilized resources. When the membrane voltage is below the synaptic threshold, these variables recover to their steady-state values of 1 and *U*, with time constants ${\tau}_{2}$ and ${\tau}_{4}$, respectively. The strength of the synapses is determined by scaling the maximal synaptic conductance by the product of the values of these variables when the presynaptic cell crosses ${V}_{\mathrm{th}}$. If the presynaptic cell fires a sequence of spikes, then the term *n* th cycle refers to the time duration between the *n* th and $n+1$st crossings of ${V}_{\mathrm{th}}$. Hence, the synaptic conductance at the start of the *n* th cycle is given by ${g}_{\mathrm{pre}\to \mathrm{post}}={\overline{g}}_{\mathrm{pre}\to \mathrm{post}}{r}_{n}{u}_{n}$, where ${r}_{n}$ and ${u}_{n}$ are the values of *r* and *u* when the presynaptic membrane potential passes synaptic threshold in the *n* th cycle (*n* is defined below).

#### 2.3.1 Steady State Synaptic Plasticity Profiles

*r*and

*u*then also reach steady states and each oscillates between a minimum and a maximum value. At steady state, when crossing the synaptic threshold, ${r}_{n}$ is at a maximum, ${r}_{\mathrm{max}}$, while ${u}_{n}$ attains its minimum, ${u}_{\mathrm{min}}$. These values can be calculated from (2.3) as

where ${t}_{a}$ and ${t}_{b}$ are the durations that the cell spends above and below ${V}_{\mathrm{th}}$, respectively.

as the synaptic strength at the time of firing of a presynaptic neuron with constant period $P={t}_{a}+{t}_{b}$. We will assume that the changes in period of the bursting neurons affect only the inter-burst duration (i.e., ${t}_{a}$ is fixed). We will henceforth refer to this relationship (2.5) as the steady-state synaptic plasticity profile.

*σ*determines the spread.

## 3 Results

We derive Poincaré maps that relate the firing times of a network of two neurons coupled with reciprocal inhibition. We assume a predetermined one-to-one firing order between the neurons. The fixed points of these maps correspond to one-to-one firings of the neurons at the steady state. It is possible to derive similar maps assuming orders of firing that are not one-to-one, but these derivations are beyond the scope of the current study. We first assume a fixed synaptic strength between the neurons in Sect. 3.1. When the synapses have a fixed strength, only the phase response information of the neurons is used to determine the network activity, as has been shown previously [8]. In Sect. 3.2 we derive maps that describe the network activity when the synapses between the neurons are plastic. We compare two cases. In one case, we assume that the synapses obey the plasticity dynamics given in Eq. (2.3). In the second case, we consider synapses that obey the corresponding steady-state values given in Eq. (2.4). The latter case results in a lower-dimensional map. In Sect. 3.3, assuming both synapses obey a steady-state plasticity profile (2.6), we examine how changes in these profiles determine the network period and relative phase relations. We find conditions for a network to be able to keep a fixed firing period but vary the relative firing phase between its neurons, and vice versa.

### 3.1 Map for Phase with Static Synaptic Strength

We start with a network of two oscillatory neurons reciprocally inhibiting each other with constant synaptic strength. We will derive a 1D map that measures the phase difference between the onset of firing of the two cells. A fixed point of the map corresponds to a $1:1$ phase-locked solution. We then derive the criteria for existence and stability of fixed points. Finally, we test the map in a network of two M–L model neurons.

*n*, respectively for A and B.

*n*as

using (3.3b) and (3.4b).

*n*can be obtained from the relation

*f*is monotone increasing in $[0,1]$ if and only if ${f}^{\prime}(\varphi )\ge 0$ on this interval where

*ϕ*is large enough (i.e., larger than the minimum point of the PRC; see Fig. 1b). For our choice of parameters this occurs when $\varphi >0.75$ (Fig. 1b) where the PRC is increasing. On the remaining interval, the expression $1-Z(\varphi )$ is ≥1. So if ${Z}^{\prime}(\varphi )\ge -1/\varphi \ge -4/3$ on $[0,0.75]$, then ${f}^{\prime}(\varphi )$ would also be positive and

*f*could then be inverted on $[0,1]$ (Fig. 4b). However, it is not possible to analytically make this estimate since we have no closed form expression for $Z(\varphi )$. We confirmed numerically though that ${Z}^{\prime}(\varphi )\ge -4/3$ in this interval, hence ${f}^{\prime}(\varphi )$ is positive on $[0,1]$. Therefore, the function

*f*can be inverted on $[0,1]$. The numerically obtained inverse function ${f}^{-1}$ is shown in Fig. 4b. Hence, the phase of cell A (3.1a) in cycle $n+1$ can be obtained from its value in cycle

*n*from

In general, the function *f* (3.12) and the map $\tilde{\Pi}$ (3.13) can be defined for networks consisting of either identical or non-identical neurons. Here we have considered only the networks of identical neurons in this section. The generalization to networks of non-identical neurons is considered below in Sect. 3.3.3.

We can now assess the existence and stability of fixed points of the maps (3.10) and (3.13). We numerically solved the map (3.10) using MATLAB to predict the activity of two identical M–L neurons coupled with reciprocal inhibition. We also numerically solved the differential equations governing the activity of the neurons using XPPAUT [20]. We let the maximal synaptic conductance $\overline{g}$ equal 0.1 and use the PRCs of the neurons obtained for this value of synaptic strength. We first find the fixed points of the map by solving the fixed point equation (3.11). The two sides of Eq. (3.11) are plotted in Fig. 4a. They intersect only at one point ${\varphi}^{\ast}=0.598$, which corresponds to the intrinsic phase of cell A (3.2a) at the steady state. The firing period of cell A can be obtained from Eq. (3.3a) evaluated at this intrinsic phase. This value is also equal to the period of B and will be referred to as the period of the coupled network (${P}_{\mathrm{st}}$). The activity phase ${\tilde{\varphi}}^{\ast}$ of cell A (3.1a) at the steady state is 0.5 and is obtained by using (3.12), corresponding to the anti-phase solution, which agrees with the simulations (not shown). In Fig. 4c, the right and left hand sides of the fixed point equation (3.11) are plotted as functions of the activity phase using (3.12). They intersect at ${\tilde{\varphi}}^{\ast}=0.5$. In Fig. 4d, we show the cobweb diagram for the map (3.13), starting with the initial condition ${\tilde{\varphi}}_{0}=0.2$ leading to convergence to the stable steady state of ${\tilde{\varphi}}^{\ast}=0.5$. For this case, the system locks in the anti-phase state because the two neurons and the two synaptic strengths are identical.

### 3.2 Maps Using Dynamic Synapses or Steady-State Synaptic Plasticity Profiles in One Synapse

In this section we derive maps to predict the network activity in the presence of synaptic plasticity. We now let the synaptic strength from cell A to cell B be constant and the strength from cell B to cell A exhibit plasticity.

When plasticity is included in the B to A synapse, the synaptic strength is no longer constant. Hence we cannot use a unique PRC for neuron A. Instead, we define a PRC as a function of two variables, where the phase at which the synapse is received and the strength of the synapse determine the response of the neuron. We denote this by ${Z}_{\mathrm{A}}(\varphi ,g)$. The PRC of neuron B is obtained for a constant synaptic strength ${\overline{g}}_{\mathrm{A}\to \mathrm{B}}$ and is denoted by ${Z}_{\mathrm{B}}(\theta )$.

- i.
changes according to the dynamics of the plasticity variables

*r*and*u*and is given by ${\overline{g}}_{\mathrm{B}\to \mathrm{A}}{r}_{n}{u}_{n}$, or, - ii.
obeys the steady-state synaptic plasticity profile ${g}_{\mathrm{B}}(P)={\overline{g}}_{\mathrm{B}\to \mathrm{A}}{r}_{\mathrm{max}}(P){u}_{\mathrm{min}}(P)$.

*n*depends on the values of the plasticity variables in this cycle. Assume that we know the values ${\varphi}_{n}$, ${r}_{n}$ and ${u}_{n}$. Then we can compute the period of neuron A in cycle

*n*using the expression

*n*as

*n*. Using Eq. (3.7) together with the above equations gives a 3D map for the evolution of the intrinsic phase of cell A (3.2a) and the synaptic plasticity variables from cell B to cell A

The first equation is the same as (3.7) except that now ${Z}_{\mathrm{A}}$ is a function of two arguments. The second and third equations are computed using (2.3) over one cycle. The complicated expressions in the exponential of both equations are the time ${Q}_{n}-{t}_{a}$ recast in terms of ${\varphi}_{n},{r}_{n},{u}_{n}$ where ${Q}_{n}$ is given in Eq. (3.15).

*n*can be found by using (3.4a) as

*n*as

Similar to Eq. (3.14), the period of neuron A is determined by ${Z}_{\mathrm{A}}$ which is a function of two variables. However, in this case the synaptic strength received by neuron A in cycle $n+1$ depends directly on the cycle length of neuron B in cycle *n*.

Hence, the map (3.16) is reduced to a 2D map for the intrinsic phase and cycle length of neuron A. A fixed point $({\varphi}^{\ast},{r}^{\ast},{u}^{\ast})$ of the 3D map (3.16) corresponds to a $1:1$ solution. This $1:1$ solution is also represented by a fixed point of the 2D map (3.21) which occurs at $({\varphi}^{\ast},{P}^{\ast})$, where ${P}^{\ast}$ is the steady-state value obtained from (3.17) at $({\varphi}^{\ast},{r}^{\ast},{u}^{\ast})$.

To assess numerically the existence and stability of the fixed points of both the 2D map (3.21) and the 3D map (3.16), consider two identical neurons coupled with asymmetric synapses. Let the synaptic strength from neuron A to B be fixed at ${\overline{g}}_{\mathrm{A}\to \mathrm{B}}=0.1$. We use parameters for the plasticity variables that yield the steady-state plasticity function ${g}_{\mathrm{B}}(P)$ with a peak at the period 169.5, as shown in Fig. 2. Denote the steady-state network period and phase of neuron A from the 3D map (case i) as ${P}_{\mathrm{dyn}}$ and ${\varphi}_{\mathrm{dyn}}$, respectively, and the corresponding values from the 2D map (case ii) as ${P}_{\mathrm{ss}}$ and ${\varphi}_{\mathrm{ss}}$. Similarly, for static coupling, denote the steady-state network period as ${P}_{\mathrm{st}}$ and phase of neuron A as ${\varphi}_{\mathrm{st}}$.

We now examine how the steady-state phase of neuron A changes with respect to changes in the intrinsic period. The phase of neuron A depends on the value of the synaptic strength received from neuron B at the steady state. This value is determined by ${Q}^{\ast}$, the steady-state firing period of neuron B, which equals the steady-state network period ${P}^{\ast}$. When this value equals ${\overline{g}}_{\mathrm{A}\to \mathrm{B}}=0.1$, then anti-phase solutions occur. This happens for two sets of coupled neurons, where the red dashed line intersects green and black curves (Figs. 6a and 6c). Between these two points, the synaptic strength received by neuron A, given by ${g}_{\mathrm{B}}({Q}^{\ast})$, is larger than ${\overline{g}}_{\mathrm{A}\to \mathrm{B}}$. Since the cells are identical, the neurons must give equal amount of response (so that their steady-state firing periods will be equal) for a steady-state solution to occur. When both synaptic strengths are equal, both neurons have steady-state phase at 0.5. However, if ${g}_{\mathrm{B}}({Q}^{\ast})>{\overline{g}}_{\mathrm{A}\to \mathrm{B}}$, then neuron A receives stronger synaptic input than neuron B. This difference can be balanced if neuron A receives this synaptic input at a phase that yields less response. As the PRCs of the neurons are decreasing around the phase 0.5, neuron A needs to phase lock at a phase smaller than 0.5. This explains why phase of neuron A decreases between these intersection points. A similar argument holds when ${g}_{\mathrm{B}}({Q}^{\ast})<{\overline{g}}_{\mathrm{A}\to \mathrm{B}}$.

The phase of neuron A reaches a minimum when the synaptic strength reaches a maximum. As can be seen in Fig. 2, the synaptic plasticity profile has its peak at 169.5. Therefore, the minimum phase of neuron A is observed at the network period 169.5 (Fig. 6c). The network period of 169.5 is obtained when two cells with intrinsic periods 141.8 are coupled (Fig. 6b).

### 3.3 Maps Using Steady-State Synaptic Plasticity Profiles in Both Directions

*n*. Equation (3.17) can still be used to obtain the intrinsic phase of neuron B (3.2b), ${\theta}_{n}$, in cycle

*n*. However, the cycle length of neuron B is now given by the equation

*n*, since the synapse from neuron A to B also has plasticity and depends on ${P}_{n}$. The cycle length

*P*and intrinsic phase

*ϕ*of neuron A in cycle $n+1$ is given by

*P*and

*ϕ*when both synapses have plasticity. In the case where the two cells are identical, ${Z}_{\mathrm{A}}(\cdot )={Z}_{\mathrm{B}}(\cdot )=Z$, this map simplifies to

We now explore whether these equations yield stable fixed points and, if so, how changes in the synaptic profiles affect the resulting phase- and period-locking of the network.

*n*are located on the

*x*–

*y*axes. These values are mapped through the surfaces ${P}_{n+1}={\Pi}_{2}({\varphi}_{n},{P}_{n})$ (Fig. 7a) and ${\varphi}_{n+1}={\Pi}_{1}({\varphi}_{n},{P}_{n})$ (Fig. 7b) to the next iteration points $({\varphi}_{n+1},{P}_{n+1})$ in cycle $n+1$. Start with the initial condition $({\varphi}_{0},{P}_{0})$ which is shown in both coordinate systems. The image of $({\varphi}_{0},{P}_{0})$ on the surface ${\varphi}_{n+1}={\Pi}_{1}({\varphi}_{n},{P}_{n})$ gives the next intrinsic phase value ${\varphi}_{1}$, and the image of $({\varphi}_{0},{P}_{0})$ on the surface ${P}_{n+1}={\Pi}_{2}({\varphi}_{n},{P}_{n})$ gives the next cycle length ${P}_{1}$ (shown by the vertical lines with one arrow). These ${\varphi}_{1}$ and ${P}_{1}$ values are located, respectively, on the

*x*and

*y*axes of both coordinate systems (shown by the inclined lines with one arrow). The point $({\varphi}_{1},{P}_{1})$ is then located on the

*x*–

*y*axes in both coordinate systems and mapped to the point $({\varphi}_{2},{P}_{2})$ by the same procedure (shown by the lines with two arrows). We are able to geometrically observe the iterations (only three shown) approach a fixed point; hence this is a generalization of cobwebbing for the 2D map.

for identical cells.

The stability of the fixed point can be examined using the Jacobian of the 2D map (3.24). If the eigenvalues of the Jacobian at the fixed point are located inside the unit circle, the fixed point is stable. For our choice of parameter values, the fixed point can be shown to be stable.

#### 3.3.1 Phase and Period Locking for Different Synaptic Plasticity Profiles

Having determined a method for calculating the steady-state network period and phase, we now determine how these quantities depend on various network parameters. For simplicity, in this section we consider identical neurons. We use the 2D map (3.24) to obtain the network phase and period when both synapses have plasticity. For comparison, we also obtain the same from the 1D map (3.10), when the synaptic strength is fixed.

In the case of identical plasticity profiles, the neurons have the same preferred periods and the values of the plasticity profiles again approach 0.075 at the tails (Fig. 9b1). This causes ${P}_{\mathrm{ss}}$ to be smaller than ${P}_{\mathrm{st}}$ for small and large intrinsic periods (Fig. 9b2). For intermediate firing periods, the opposite holds. In contrast to the almost linear change in ${P}_{\mathrm{st}}$, ${P}_{\mathrm{ss}}$ changes nonlinearly as a function of the intrinsic periods. Also, in contrast to the nonlinear change in ${P}_{\mathrm{ss}}$, the phase of neuron A is fixed at 0.5, because both the neurons and their plasticity profiles are identical (Fig. 9b3). Hence, depending on the choice of plasticity profiles, the network coupled with synaptic plasticity can have the same period but different relative phases (Fig. 9a1–a3), or the same phases but different periods compared to the network coupled with static synapses (Fig. 9b1–b3).

#### 3.3.2 Conditions for Phase or Period Constancy

Short-term synaptic plasticity profiles are subject to change by neuromodulation and other long-term modifications [21]. In the previous section, we showed that as the synaptic plasticity profile changes, the network can maintain the network period or the relative activity phases among the network neurons. In this section, we examine the conditions on the steady-state synaptic plasticity profiles that would allow the network to maintain either a constant period or a constant phase.

*P*and synaptic preferred periods ${P}_{\mathrm{A}}$ and ${P}_{\mathrm{B}}$ as

*P*and $\tilde{\varphi}$. Using the Implicit Function Theorem, the condition that needs to be satisfied is $det({D}_{{P}_{\mathrm{A}},{P}_{\mathrm{B}}}F)\ne 0$ at $({P}_{\mathrm{A}}^{\ast},{P}_{\mathrm{B}}^{\ast},{\tilde{\varphi}}^{\ast},{P}^{\ast})$ where

One condition for the determinant to be nonzero is $\partial Z/\partial y(x,y){|}_{({P}_{\mathrm{A}}^{\ast},{P}_{\mathrm{B}}^{\ast},{\tilde{\varphi}}^{\ast},{P}^{\ast})}\ne 0$; that is, the response of the neuron to perturbations should change with the change in the strength of the perturbation. This is a standard assumption on phase response curves with small perturbations. The other two conditions to be satisfied are $\partial {g}_{\mathrm{A}}/\partial {P}_{\mathrm{A}}{|}_{({P}_{\mathrm{A}}^{\ast},{P}_{\mathrm{B}}^{\ast},{\tilde{\varphi}}^{\ast},{P}^{\ast})}\ne 0$ and $\partial {g}_{\mathrm{B}}/\partial {P}_{\mathrm{B}}{|}_{({P}_{\mathrm{A}}^{\ast},{P}_{\mathrm{B}}^{\ast},{\tilde{\varphi}}^{\ast},{P}^{\ast})}\ne 0$, which, upon using Eq. (2.6), are equivalent to ${P}_{\mathrm{A}}\ne {P}^{\ast}$ and ${P}_{\mathrm{B}}\ne {P}^{\ast}$, respectively. In other words, the network period should be different from the synaptic preferred periods.

Under these three conditions, the Implicit Function Theorem guarantees that ${P}_{\mathrm{A}}$ and ${P}_{\mathrm{B}}$ can be expressed in terms of *ϕ* and *P* near $({P}_{\mathrm{A}}^{\ast},{P}_{\mathrm{B}}^{\ast},{\tilde{\varphi}}^{\ast},{P}^{\ast})$. More precisely, there are neighborhoods *U* of $({\tilde{\varphi}}^{\ast},{P}^{\ast})$ and *W* of $({P}_{\mathrm{A}}^{\ast},{P}_{\mathrm{B}}^{\ast})$ such that, for each $(\tilde{\varphi},P)\in U$, there exists a unique $({P}_{\mathrm{A}},{P}_{\mathrm{B}})\in W$ such that $F({P}_{\mathrm{A}},{P}_{\mathrm{B}},\tilde{\varphi},P)=F({P}_{\mathrm{A}}(\tilde{\varphi},P),{P}_{\mathrm{B}}(\tilde{\varphi},P),\tilde{\varphi},P)=0$. Hence, there is a unique function $h=({h}_{1},{h}_{2}):U\to W$ such that $F({h}_{1}(\tilde{\varphi},P),{h}_{2}(\tilde{\varphi},P),\tilde{\varphi},P)=0$ for every $(\tilde{\varphi},P)\in U$.

We can interpret this result in two ways. First, around the fixed point $({P}_{\mathrm{A}}^{\ast},{P}_{\mathrm{B}}^{\ast},{\tilde{\varphi}}^{\ast},{P}^{\ast})$, we can choose $({\tilde{\varphi}}^{\prime},{P}^{\ast})$ such that ${P}^{\ast}$ is fixed and ${\tilde{\varphi}}^{\prime}\ne {\tilde{\varphi}}^{\ast}$, for which there exist $({P}_{{\mathrm{A}}^{\prime}},{P}_{{\mathrm{B}}^{\prime}})$ that satisfy the fixed point equations (3.26). Hence, for a specific ${P}^{\ast}$, around a point with a phase ${\tilde{\varphi}}^{\prime}$, there exist synaptic preferred periods ${P}_{{\mathrm{A}}^{\prime}}$ and ${P}_{{\mathrm{B}}^{\prime}}$ that enable the network to stay on the level set of ${P}^{\ast}$, while setting the phase equal to a new value ${\tilde{\varphi}}^{\prime}$. In other words, it is possible to keep the network period constant and set the network phase to a new value by changing the synaptic plasticity profiles of the network neurons.

The second interpretation is that, around the fixed point $({P}_{\mathrm{A}}^{\ast},{P}_{\mathrm{B}}^{\ast},{\tilde{\varphi}}^{\ast},{P}^{\ast})$, we can choose a $({\tilde{\varphi}}^{\ast},{P}^{\prime})$ such that ${\tilde{\varphi}}^{\ast}$ is fixed and ${P}^{\prime}\ne {P}^{\ast}$, and can find $({P}_{{\mathrm{A}}^{\prime}},{P}_{{\mathrm{B}}^{\prime}})$ that satisfy the fixed point equations (3.26). This enables the network to stay on the level set for a specific ${\tilde{\varphi}}^{\ast}$, while changing the network period to a new value ${P}^{\prime}$.

#### 3.3.3 Networks of Non-identical Neurons

Note that the maps continue to give good predictions when the neurons are not necessarily identical. The difference between the simulations (filled circles) and the map predictions (open circles) is indistinguishable in most cases. The diagonal corresponds to coupling of identical neurons. Moving away from the diagonal, the difference between the intrinsic periods of the neurons increases and eventually prevents the neurons to phase lock in a $1:1$ manner because the fixed point equation (3.8) is not satisfied anymore. These are the limits of the region shown in Fig. 11. Observe that the limits determined by the map and the simulations overlap except at one single case shown only by an open circle in Figs. 11c and 11d. Here, the map predicts that a $1:1$ solution exists, while the simulation does not converge to that. In this case, the simulation shows that the firing order between the neurons is not preserved which violates the $1:1$ firing assumption of the map.

The phase of neuron A equals 0.5 on the diagonal in the static coupling case (Fig. 11a). It decreases (resp. increases) linearly as ${Q}_{0}$ moves down (resp. up) from the diagonal. This behavior can be predicted by analyzing Eq. (3.8). In the identical network, where ${P}_{0}={Q}_{0}$, the activity phases (${\tilde{\varphi}}^{\ast}={\tilde{\theta}}^{\ast}=0.5$), and the intrinsic phases (${\varphi}^{\ast}={\theta}^{\ast}=0.598$) of the two neurons are equal and hence ${Z}_{\mathrm{A}}({\varphi}^{\ast})={Z}_{\mathrm{B}}({\theta}^{\ast})$. If the solution is perturbed such that ${P}_{0}>{Q}_{0}$, then the response of neuron A to synaptic inputs from neuron B must be smaller than the response of neuron B for the Eq. (3.8) to be satisfied. The PRC of the neurons has a negative slope at this intrinsic phase ${\varphi}^{\ast}$ (Fig. 1b). So, the intrinsic phase *ϕ* of neuron A in the perturbed solution must be smaller than ${\varphi}^{\ast}$ for ${Z}_{\mathrm{A}}(\varphi )$ to be smaller than ${Z}_{\mathrm{A}}({\varphi}^{\ast})$. As the function (3.12) relating *ϕ* and $\tilde{\varphi}$ is monotone increasing, the activity phase $\tilde{\varphi}$ of neuron A in the perturbed solution must also be smaller than ${\tilde{\varphi}}^{\ast}$. Hence, as the difference ${P}_{0}-{Q}_{0}$ increases (resp. decreases), the phase of neuron A decreases (resp. increases). The period of the network increases linearly as the intrinsic periods increase in the static coupling case (Fig. 11b). Due to symmetry in the synaptic strengths, the distribution of the period is symmetric with respect to the diagonal.

When the synapses are plastic, some $1:1$ phase-locked solutions that existed with static coupling no longer exist, while new solutions may emerge (Figs. 11c and 11d). Due to asymmetry in the synaptic plasticity profiles, the upper bound for the difference in intrinsic periods that allow a $1:1$ phase-locked solution varies. This can be seen by comparing the circles in the top row and rightmost column of Figs. 11c and 11d. At the right top corner, ${P}_{0}={Q}_{0}=181$, and the network has an anti-phase solution. If ${Q}_{0}$ is fixed, while ${P}_{0}$ decreases, the network continues to phase lock in a $1:1$ solution for ${P}_{0}\ge 152.1$. On the other hand, if ${P}_{0}$ is fixed, while ${Q}_{0}$ decreases, then the network phase locks in a $1:1$ solution only when ${Q}_{0}\ge 174.8$. Although the absolute difference between the intrinsic periods are equal, different plasticity profiles causes convergence in one case but not the other. This can be understood by considering (3.25). For the identical cell case where ${P}_{0}={Q}_{0}=181$, the network period is equal to ${P}^{\ast}=219.5$. Due to the selection of the plasticity profiles, ${g}_{\mathrm{A}}({P}^{\ast})<{g}_{\mathrm{B}}({P}^{\ast})$, since ${P}^{\ast}$ is close to ${P}_{\mathrm{B}}=190$ than it is to ${P}_{\mathrm{A}}=150$. As a result, neuron A receives stronger synaptic input from neuron B at the steady state (as ${g}_{\mathrm{B}}({P}^{\ast})$ determines ${g}_{\mathrm{B}\to \mathrm{A}}$). The firing periods of both neurons must be equal at the fixed point. This is only possible if neuron B receives synaptic input at a phase that yields a larger response than that of neuron A. Hence, although the neurons are identical, the difference in their plasticity profiles causes a phase-locking solution different from anti-phase. Assume now that ${Q}_{0}>{P}_{0}$. Then the relation ${g}_{\mathrm{A}}(P)<{g}_{\mathrm{B}}(P)$ will still hold as *P* will stay close to ${P}^{\ast}$. In this case, the synaptic strength received by neuron A will be larger, while its intrinsic period will be smaller than that of B. These two opposing effects will let the network continue having a solution until the difference between the intrinsic periods are too large to be compensated for and (3.25) are not satisfied. On the other hand, if the symmetric solution is perturbed such that ${P}_{0}>{Q}_{0}$, then the synaptic strength received by neuron A and its intrinsic period will both be larger than those of neuron B. The phase of neuron B must increase further and yield a larger response to compensate for these adding effects. But when the PRC reaches a maximum in absolute value and starts to decrease, there would be no phase value that would compensate for these effects and the network will not be able to have a $1:1$ solution. This explains why the limits of the regions in the case with synaptic plasticity are not symmetrical.

In general, whether (3.25) are satisfied or not depends on the intrinsic periods ${P}_{0}$, ${Q}_{0}$ and the values of the PRCs as in the static map case. But in this case the values of the PRCs are also determined by two factors, the phase of inhibition received, and its strength—which is determined by the network period. Hence, the phase of neuron A is a determined both by the interaction of intrinsic periods and the plasticity profiles. This is also responsible for the nonlinearity in the distribution of phase. The level curves of phase are nonlinear in the case with synaptic plasticity as opposed to the linear level curves in the static coupling case.

## 4 Discussion

In the analysis of an oscillatory network, the steady-state activity of the network can often be reduced to the study of a return map. The advantage of using maps is that it often allows the network dynamics to be understood by tracking empirically observable characteristics such as period and phase. Here, we derive such a map for a two-cell network coupled with inhibitory synapses with the goal of understanding how short-term synaptic plasticity and other factors determine the network period and the relative activity phase of the two neurons. Our results show that the information on the network period and phase can be obtained using maps that keep track of observable network variables such as the intrinsic periods of the neurons involved, their phase response curves and the synaptic plasticity profiles: relationships describing how the synaptic strength depends on input frequency. These variables can be readily determined experimentally with “feed-forward” measurements where the input is controlled by the experimenter and the output is measured. For example, the strength of a synapse can be measured at all frequencies simply by driving the presynaptic neuron at different rates and measuring the postsynaptic current. In fact, the current study was motivated by our experimental measurements of these types of network variables in the crab stomatogastric pyloric network [22–24].

There are several prior works that utilize PRCs and map-based techniques to understand phase locking [1–13]. Of particular interest is the result of Cui et al. [5] who use a functional PRC (fPRC) that is calculated from actual experimental measurements of Aplysia pacemaker neurons. Cui et al. show that the fPRC differs from the single pulse PRC (as was used in this paper) due to accommodation of the pacemaker neurons. They then go on to use the fPRC to study phase-locking in a coupled network by deriving a map that encodes how a neuron responds to a period input that arrives a fixed time after the firing of the cell. By linearizing about a fixed point of their 1D map, they find conditions for the existence and stability of $1:1$ phase-locked solutions. Their predictions from the fPRC method are better matched to simulations than predictions from a conventional single-pulse PRC. Importantly, their fPRC methods do not depend on the exact shape of the PRC but rather on the effect on the cycle period based on the time the input was given. This is a statistic that is easily found in experiments. Moreover, their results are obtained from combining feed-forward processes as opposed to directly studying a feedback map, which they call open-looped versus closed-looped.

Our results complement those of Cui et al. in the sense that we relate cycle-to-cycle changes in the period independent of how those changes arise, allowing us to also use experimentally obtainable information to derive the maps. Our maps are also based on assumptions that are consistent with Cui et al.’s assumption that the closed-loop behavior of a system can be predicted by knowing the open-looped behavior of some of its components. Our results extend those of Cui et al. and other prior works in that we allow the timing of inputs to vary on a cycle by cycle basis that is determined by the synaptic plasticity profile of the presynaptic neuron. This results in a higher-dimensional map arising by specifically considering the dynamics of synaptic facilitation and depression on a cycle by cycle basis. This yields a 3D map when plasticity is present only in one direction of the two-cell network, or a 5D map if present in both directions. When we used the steady-state synaptic plasticity profile, both cases reduce to a 2D map. For this 2D map, we derived a geometric method that generalizes cobwebbing in a 1D map to allow us to study the existence and stability of fixed points. For a generic 1D map, $\Pi (x)$, the intersection of the curve $y=\Pi (x)$ with the curve $y=x$, and the slope at that point, determine existence and stability of the fixed point. In our generalized 2D case, given maps ${\Pi}_{1}(x,y)$ and ${\Pi}_{2}(x,y)$, it is the intersection of these surfaces with appropriate planes that yield two curves. It is the intersection of the projection of these two curves onto a common plane that determines existence of the fixed point. Stability is more complicated than just checking the slopes at the point of intersection. We showed how it could depend on both the PRC and the synaptic plasticity profile.

In this study, we considered a general form of short-term synaptic plasticity which is a combination of short-term facilitation and depression. We modeled such a synapse using an ad hoc model as described previously [16]. The advantage of this model is that the extent to which facilitation or depression is a dominant factor can be simply determined by changing the model parameters. Our analysis progressed through a network of two neurons with static synapses, the same network but with one synapse having plasticity and finally with both synapses showing plasticity. The analysis of a two-cell network with static synapses yields a 1D map [6, 8]. Including synaptic plasticity increases the dimension of the map because variables underlying synaptic dynamics must be tracked as well. The change in synaptic strength due to the plasticity means that the PRCs of the neurons also change. Our analysis shows that these higher-dimensional maps can accurately predict the steady-state phase and period of the network, as seen in comparisons with numerical simulations of the underlying ODEs.

In experimental measurements, synaptic plasticity profiles are often measured using repetitive input pulses or waveforms and reported at steady state, i.e., the steady-state strength of the synapse is known for each stimulation frequency [23, 25, 26]. In most cases, the mechanisms that underlie these synaptic dynamics are unknown and it is therefore impossible to track how synaptic strength changes as a function of frequency on a cycle-to-cycle basis. One of the interesting findings from our work is that the prediction of the higher-dimensional map obtained when using dynamics of the synapse is the same as a lower-dimensional map that uses only the steady-state plasticity profile. In other words, the network output is dependent on the steady-state strength independent of the mechanisms through which this synaptic strength is actually generated. In turn, this allows an experimentalist to understand the effects of, say a synaptic neuromodulator, on the network output simply by understanding the effect on a single component such as the synaptic plasticity profile.

The results of our maps help us understand the role of synaptic dynamics in determining the relative phase between two neurons in an oscillatory network. For example, neurons in the crustacean pyloric oscillatory network, involve multiple reciprocally inhibitory connections. Pyloric oscillations are quite stable in individual preparations and are generated by a pacemaker group of neurons (AB/PD) which make reciprocally inhibitory connections with a single follower neuron, LP. The analysis of this reciprocally inhibitory network provided the motivation for the current study. As in other rhythmic motor networks, the pyloric network neurons maintain a constant phase relationship even when these phases are measured in different animals [27]. Surprisingly this tight phase relationship is maintained despite a large variability in the pyloric cycle period (1–2.5 s) across preparations. In fact, different preparations differ both in the intrinsic periods of the neurons involved as well as the synaptic plasticity profiles. The results of the current study indicate that the pyloric network could maintain constant phase relationships, even in different animals, by tuning the synaptic plasticity profiles along the level sets of phase (Fig. 10). Alternatively, if the relative activity phases of the neurons involved in producing the network oscillations are not an essential component of the network output, but the network must maintain a constant period, the maps we have derived can be used to establish the relationships that could produce a constant frequency output. These are plausible strategies for all rhythmic motor networks in which the output is tightly constrained by the proper phase of muscle movements to produce meaningful behavior.

An interesting implication of our results is that if the network period coincides with the synaptic preferred periods, it is not possible to uniquely prescribe the synaptic profiles in terms of the network period and the relative phase of the neurons (Eq. (3.27)). If the level sets of phase, described in Fig. 10, provide a unique rule for the network to tune its synaptic plasticity profiles for phase maintenance, then the network period should avoid the synaptic preferred periods. Additionally, by avoiding the periods at which the synaptic strengths are maximal, the network can operate with a larger degree of flexibility and perhaps more efficiently. This is in fact the case for the synapses between the AB/PD pacemaker neurons and the follower LP neuron in the crustacean pyloric network. The network period is around 1–2.5 s, in a range of values that is larger than the preferred periods of the synapses (∼0.5 Hz) [23]. Hence, our findings give an insight for this experimentally observed fact.

In conclusion, we have shown that the frequency-dependent information on synapses can be combined with the PRCs of oscillatory neurons to predict the activity period and phases of a coupled network using maps derived from empirically observable relationships. It is plausible that a similar approach can be used whenever there is frequency-dependent information about the network components to construct maps that predict the activity of an oscillatory network, even when the synapses include excitatory connections or obey different plasticity profiles. In relationship to the crustacean pyloric network that motivated this study, current experimental work in our lab involves measuring the changes in the synaptic plasticity profiles and the neuronal PRCs in the presence of different neuromodulators to see whether the maps derived here can predict how the network output changes in the presence of these modulators.

## Declarations

### Acknowledgements

This work was supported by NSF DMS 1122291 and NIH MH060605.

## Authors’ Affiliations

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