The idea behind our proof can be explained graphically. We assume that nodes cannot oscillate due to their intrinsic activity and a fixed external input only drives them to a non-zero activity which does not change over time. Therefore, they need input from their pre-synaptic (upstream) nodes to change their state in a periodic manner to generate oscillations. In such a network, if we perturb the node i with a pulse-like input, it is necessary that the perturbation travels through the network and returns to the node i with a 180° phase shift (i.e. with an inverted sign). Otherwise, the perturbation dies out and each node returns to a state imposed by its external input.

In a network without directed cycles, it is possible to sort the nodes into smaller groups where nodes do not connect to each other (Figure 2a). That is, a network with no directed cycles, can be rendered as a feed-forward network in which the network response by definition does not return to the node (or group) that was perturbed. Such a network can only oscillate when the intrinsic dynamics of individual nodes allow for oscillatory dynamics.

Figure 2

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However, having a directed cycle is no guarantee of oscillations because network activity must return to the starting node with a 180° phase shift. This requirement puts a constraint on the number of inhibitory connections in the cycles. When we assume that there are no delays (or the delay is constant) in the connections, excitatory connections do not introduce any phase shift, however, inhibitory connections shift the phase by 180° (in the simplest case invert the sign of the perturbation). Given this, when a cycle has an even number of inhibitory connections the cycle cannot exhibit oscillations (Figure 2b,top). However, replacing an inhibitory connection by an excitatory one can render this cycle with an ability to oscillate (Figure 2b, bottom). Therefore, odd number of inhibitory appears to be necessary for oscillation to emerge.

To test the validity of our theorems in more realistic biological neuronal networks, we numerically simulated the dynamics of the Wilson-Cowan model. Specifically, we investigated the role of synaptic transmission delays, synaptic weights, external inputs and self-connection in shaping the oscillations when the network has directed cycles. In particular, we focused on two networks: the III motif with three inhibitory nodes (odd inhibitory links) and the EII motif with one excitatory and two inhibitory nodes (even inhibitory links).

Our numerical simulation showed that for a wide range of parameters (synaptic delays, synaptic weights, external input and self-inhibition), while III network showed oscillations, EII network only resulted in transient oscillations (Figure 3, Figure 3—figure supplements 1 and 2). The oscillation frequency however depended on the exact value of the synaptic delays, synaptic weights and external inputs. For instance, increasing the synaptic delay reduced the oscillation frequency (Figure 3b). Synaptic delays play a more important role in shaping the oscillations in an EI type network (see Figure 3—figure supplement 3, Appendix 2—table 1). In networks with even number of inhibitory connections (e.g. EII, EEE, II), synaptic delays are the sole mechanism for initiating oscillations, however, unless delays are precisely tuned such oscillations will remain be transient (see Figure 3—figure supplement 2). The effect of increasing the synaptic strength was contingent on the external inputs. In general, increasing the synaptic strength resulted in a reduction in the oscillation frequency (Figure 3c). Next, oscillation frequency changed in a non-monotonic fashion as a function of external input irrespective of the choice of other parameters (Figure 3d). Typically, a mid-range input strength resulted in maximum oscillation frequency. Finally, increasing the self-connection of nodes increased the oscillation frequency but beyond a certain self-connection the node was completely silenced and it changed the network topology and oscillations disappeared (Figure 3e).

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Overall, these results are consistent with our rule that odd number of inhibitory nodes and strong enough connections are necessary to induce oscillations in a directed cycle. The actual frequency of oscillations depends on specific network parameters.

Next, we use our theorem to explain recent experimental observations about the mechanisms underlying the emergence of oscillations in the BG. Emergence of 15–30  Hz (beta band) oscillations in the cortico-basal ganglia (CBG) network is an ubiquitous feature of PD Raz et al., 2000; Bergman et al., 1998; Sharott et al., 2014; Neumann et al., 2016. Based on their connectivity and activity subthalamic nucleus (STN) and the globus pallidus externa (GPe) subnetwork has emerged as the most likely generator of beta oscillations Plenz and Kital, 1999; Bevan et al., 2002. The STN-GPe subnetwork becomes oscillatory when their mutual connectivity is altered Terman et al., 2002; Holgado et al., 2010 or neurons become bursty Tachibana et al., 2011; Bahuguna et al., 2020 or striatal inputs to GPe increase Kumar et al., 2011; Mirzaei et al., 2017; Sharott et al., 2017; Chakravarty et al., 2022. However, oscillations might also be generated by the striatum McCarthy et al., 2011, by the interaction between the direct and hyperdirect pathways Leblois et al., 2006 and even by cortical networks that project to the BG (Brittain and Brown, 2014). Recently, Crompe et al., 2020 used optogenetic manipulations to shed light on the mechanisms underlying oscillation generation in PD. They showed that GPe is essential to generate beta band oscillations while motor cortex and STN are not. These experiments force us to rethink the mechanisms by which beta band oscillations are generated in the CBG network.

To better understand when GPe and/or STN are essential for beta band oscillations, we identified the network motifs which fulfill the odd inhibitory cycle rule. For this analysis, we excluded D1 SPNs because they have a very low firing rate in the PD condition Sharott et al., 2017. In addition, cortex is assumed as a single node in the CBG network.

The CBG network can be partitioned into 246 subnetworks with 2, 3, 4, 5, 6, or 7 nodes (see Figure 4—figure supplements 1–6). Among these partitions, there are five loops (or cycles) in the CBG network with one or three inhibitory projections: Proto-STN, STN-GPi-Th-cortex, STN-Arky-D2-Proto, Proto-Arky-D2, Proto-FSN-D2, and Proto-GPi-Th-Cortex-D2 (Figure 4a). One or more of these 6 loops appeared in 96 (out of 246) subnetworks of CBG (see Figure 4b, colors indicate different loops). Larger subnetworks consisting of 5 and 6 nodes have multiple smaller subnetworks (with 2 or 3 nodes) that can generate oscillations (boxes with multiple colors in Figure 4b).

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Based on our odd inhibitory cycles in BG, we found three oscillatory subnetworks which do not involve the STN (Figure 4a, cyan, green and purple subnetworks). However, each of these oscillatory subnetworks involves Prototypical neurons (from the GPe) which receive excitatory input from STN. Therefore, it is not clear whether inhibition of STN can affect oscillations or not. To address this question, we first simulated the dynamics of a four-node motif (Figure 4c top) using the Wilson-Cowan type model (see Methods). In this subnetwork, we have three cycles: Proto-STN loop with one inhibitory connection, Proto-STN-Arky-D2 loop with three inhibitory connections and Proto-Arky-D2 with three inhibitory connections.

We systematically varied external inputs to the STN and D2-SPNs and measured the frequency of oscillations (see Methods). We found that for weak inputs to the D2-SPNs, the Proto-STN subnetwork generated oscillations for weak positive input (Figure 4c bottom). However, as the input to D2-SPNs increased, the oscillation frequency decreased and oscillations were observed even for very strong drive to STN (Figure 4c bottom). That is, in this model, both Proto-STN and Proto-D2-Arky subnetworks compete for oscillations, which subnetwork wins depending on their inputs. To disentangle the oscillations of each of these two subnetworks, we performed ‘lesion’ experiments in our model (see Methods). These experiments also mimicked lesions performed in non-human primates Tachibana et al., 2011.

When we removed the D2-SPN to Proto projections, the network could oscillate but only because of the Proto-STN subnetwork (Figure 4d). In this setting, we get relatively high-frequency beta band oscillations but only for a small range of excitatory inputs to the STN (Figure 4d, bottom). In this setting, inhibition of STN would certainly abolish any oscillation. Next, we removed the STN output (equivalent to inhibition of STN), the Proto-D2-Arky subnetwork generated oscillations for weak positive inputs to the D2-SPNs (Figure 4e, bottom). Note that unlike in Figure 4c, here we injected additional input to Proto to compensate for the loss of excitatory input from STN and to ensure that it had sufficient baseline activity. The frequency of Proto-D2-Arky oscillations was smaller than that observed for the Proto-STN subnetwork because the former involves a three synapses loop. However, as we have shown earlier, frequency of oscillation can be changed by scaling the connection weights or external inputs (Figure 3). Overall, these results suggest that, in principle, it is possible for CBG network to oscillate even when STN is removed from the network.

Thus far we have only illustrated the validity of our theorems in a firing rate-based model. To be of any practical value to brain science, it is important to check whether our theorems can also help in a network with spiking neurons. To this end, we simulated the two subnetworks with 3 inhibitory connections: Proto-D2-FSN and Proto-D2-Arky (see Methods). These subnetworks were simulated using a previous model of BG with spiking neurons Chakravarty et al., 2022.

The subnetworks Proto, Arky, D2-SPN, FSN have very little recurrent connectivity to oscillate on their own. We provided Poisson type external input. All neurons in a subnetwork received the same input rate but a different realization of the Poisson process. Both Proto-D2-FSN (Figure 5a) and Proto-D2-Arky (Figure 5b) subnetworks showed $\beta$-band oscillations. In the loop Proto-D2-FSN, D2-SPN neurons have a relatively high firing rate. This could be a criterion to exclude this loop as a potential contributor to the beta oscillations.

Figure 5

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Next, we mimicked the STN inhibition experiments performed by Crompe et al., 2020 in our model. To this end, we simulated the dynamics of BG network excluding D1-SPNs (because of their low firing rate in PD condition) and FSN (because with FSNs in the oscillation loop, D2-SPNs may have non-physiological firing rates). In this reduced model of BG, we changed inputs to operate in a mode where either Proto-D2-Arky (Figure 5c) or Proto-STN (Figure 5d) loop was generating the oscillations. In both cases, we systematically increased the inhibition of STN neurons.

In the Proto-D2-Arky mode, as we inhibited STN neurons, firing rate of the Proto neurons decreased and oscillations in STN population diminished but Proto neurons showed clear beta band oscillations (Figure 5c). By contrast, and as expected when STN-Proto loop was generating the oscillations, increasing the STN inhibition abolished the oscillations in both STN and Proto neurons (Figure 5d). In STN-Proto loop, when STN is inhibited, there is no cycle left in the network and therefore oscillations diminished, whereas the Proto-D2-Arky loop remains unaffected by the STN inhibition (except for a change in the firing rate of the Proto neurons). As shown in Figure 4c, whether oscillations are generated by the Proto-D2-Arky or STN-Proto loop depends on the relative input to the D2 or STN neurons. So it is possible that in rodents, D2-SPN have stronger input from the cortex than STN and therefore, oscillations survive despite near complete inhibition of STN.

Previous models suggest that when we excite the excitatory node or inhibit the inhibitory node oscillations can emerge and strengthen Kumar et al., 2011; Ledoux and Brunel, 2011. By contrast, when we inhibit the excitatory node or excite the inhibitory node, oscillations are quenched. This can be summarised as the ’Oscillations Sign Rule’. Let us label the excitatory population as positive and inhibitory as negative. Let us also label excitatory inputs as positive and inhibitory inputs as negative. Now if we multiple the sign of the node and sign of the stimulation, we can comment on the fate of oscillations in a qualitative manner. For example, inhibition of inhibitory nodes would be − × − = + that is oscillations should be increased and when we inhibit excitatory nodes, it would be − × + = − that is oscillations should be decreased. The ’Oscillation sign rule’ scales to larger network with more nodes. With the ’Odd Cycle Rule’ as we have shown we can comment on whether a directed cycle will oscillate or not from the count of inhibitory links. When we combine the ’Oscillations Sign Rule’ with the ’Odd Cycle Rule’ we can get a more complete qualitative picture of whether a stimulating a node in a network will generate oscillations or not.

In our proof we have assumed that nodes follow rather simple dynamics and have a threshold-linear transfer-function. In reality nodes in physical, chemical and biological systems can have more complex dynamics. For instance, biological neurons have the property of spike frequency adaptation or rebound spiking. Similarly, synapses in the brain can increase or decrease their weights based on the recent history of inputs which is referred to as the short-term-facilitation or short-term-depression of synapses Stevens and Wang, 1995. Such biological properties can be absorbed in the network structure in the form of an extra inhibitory or excitatory connection. When nodes can oscillate given their intrinsic dynamics then the question becomes more about whether a network structure can propagate oscillations to other nodes.

We applied our results to understand the mechanisms underlying the emergence of PD-related pathological oscillations in the BG. Given that there are 8 key neuron populations in the BG, we enumerated 238 possible directed cycles. From 2-node-motifs to 6-node-motifs, our odd cycle rule identified 88 potential directed cycles that can generate oscillations. Among these, 81 cycles feature GPe (either Proto or Arky type or both) and 66 feature STN. Which specific cycle underlies oscillations depends on the exact input structure. For instance, when input to STN is higher than the D2 neurons, the STN-GPe network generates oscillations. But when inputs to D2 neurons are stronger, the D2-Proto-Arky cycle can become the oscillator. That is, STN is not necessary to generate oscillations in the BG. Our results also suggest that besides focusing on the network connectivity, we should also estimate the inputs to different nodes in order to pinpoint the key nodes underlying the PD-related pathological oscillations - that would be the way to reconcile the recent findings of Crompe et al., 2020 with previous results.

We would like to note that recently, Ortone et al., 2023 and Azizpour Lindi et al., 2023 have provided a more quantitative explanation of the experimental observation by Crompe et al., 2020. They also eventually resort to the D2-Proto-Arky cycle to explain BG oscillations in the absence of STN. Our work provides the necessary theory to explain their results.

In this work, we have used the odd cycle rule to study oscillations in BG. However, oscillatory dynamics and the odd cycle rule show up in many chemical, biological and even social systems such as neuronal networks Bel et al., 2021, psychological networks Greenwald et al., 2002, social and political networks Leskovec et al., 2010; Milo et al., 2004; Heider, 1946; Cartwright and Harary, 1956, resting-state networks in autism Moradimanesh et al., 2021 and gene networks Farcot and Gouzé, 2010; Allahyari et al., 2022. In fact, originally Thomas’ conjecture Thomas, 1980 about the structural conditions for oscillations was made for gene regulatory networks. Therefore, we think that insights obtained from our analytical work can be extended to many other chemical, biological and social networks. It would be interesting to check to what extent our prediction of quenching oscillation by exciting the excitatory nodes holds in other systems besides biological neuronal networks.

In the firing rate-based models, we reduced each CBG subnetwork to a single node. To describe firing rate dynamics of such a node, we used the classic Wilson-Cowan model Wilson and Cowan, 1972

where $r_i(t)$ is the firing rate of the ith node, $\tau$ is the time constant of the population activity, $n$ is the number of nodes (or subnetworks), $w_{ij}$ is the strength of connection from node $j$ to $i$, and $I_i^{ext}$ is the external input to the population. $F$ is a nonlinear activation function relating output firing rate to input, given by

where the parameter θ is the position of the inflection point of the sigmoid, and a/4 is the slope at θ. Here, τ, θ, and a are set as 20, 1.5, and 3. Other parameters of the model varied with each simulation. The simulation specific parameters for Figure 3 are shown in Tables 1 and 2 and for Figure 4 are shown in Table 3, respectively.

The BG network with spiking neurons was taken from a previous model by Chakravarty et al., 2022. Here, we describe the model briefly and for details we refer the reader to the paper by Chakravarty et al., 2022.

Each neuron in each sub-network of the BG received external input in the form of excitatory Poisson-type spike trains. This input was provided to achieve a physiological level of spiking activity in the network. For more details please see Chakravarty et al., 2022. Briefly, the external input was modelled as injection of Poisson spike-train for a brief period of time by using the inhomogeneous_poisson_generator device in NEST. The strength of input stimulation can be controlled by varying the amplitude of the EPSP from the injected spike train.

We set a subnetwork of BG to study how STN inhibition affects oscillation when different motifs dominate the system. The connections and external inputs to each neuron in Figure 5c and 5d are shown in Appendix 2—tables 7 and 8. To simulate the increasing inhibition to STN, the external input to STN was reduced from 1 pA to –99 pA in Figure 5c and from 30 pA to –50 pA in Figure 5d.

The estimate of oscillation frequency of the firing rate-based model was done using the power spectral density calculated by pwelch function of MATLAB. The spiking activity of all the neurons in a sub-population were pooled and binned (rectangular bins, bin width = 0.1ms). The spectrum of spiking activity was then calculated for the binned activity using pwelch function of MATLAB.

Wilson-Cowan type firing rate-based model was simulated using Matlab. All the relevant differential equations were integrated using Euler method with a time step of 0.01ms. The network of spiking neurons was simulated in Python 3.0 with the simulator NEST 2.20 Fardet et al., 2020. During the simulation, differential equations of BG neurons were integrated using Runga–Kutta method with a time step of 0.1ms.

The code to simulate key results is available at https://github.com/jiezang97/Code-for-Structural-constraints-on-the-emergence-of-oscillations, copy archived at Zang, 2024.

In this section, we prove theorems 1 and 2 using similar notations as Curto et al., 2012.

Let $x_0\in {ℝ}^n$ and denote by

a solution of the threshold-linear network dynamical system TLN($W,b$) (Equation 1) with $x^{x_0}(0)=x_0$. A fixed point $x^{*}\in {ℝ}^n$ of TLN($W,b$) is defined as a point such that for all ${\displaystyle t\ge 0}$,

Formally, such a point satisfies for all $i\in [n]$,

The support of a fixed point $x^{*}$,

is the subset of active nodes of $x^{*}$.

In what follows, we consider a subset $\sigma \subset [n]$ and denote by $\overline{\sigma }=[n]\setminus \sigma$. For a given $n$ by $n$ matrix $W$ and $\sigma ,\tilde{\sigma }\subset [n]$, we note

Moreover, we denote by $(I_{\sigma }-W_{\sigma };b_{\sigma },i)$ the matrix $I_{\sigma }-W_{\sigma }$ with the $i^{th}$ column replaced by $b_{\sigma }$.

Theorem 1 says that directed cycles are necessary to help a network to oscillate based on structural conditions.

Proof of Theorem 1. If ${\displaystyle G}$ does not contain any directed cycle, it is called an acyclic digraph. From Proposition 2.1.3 in Bang-Jensen and Gutin, 2009, we can always group the nodes into an acycling (or topological) ordering as follows (see Figure 2a for an illustration). There exists $k\in [n]$ and a partition ${\sigma }_1,{\sigma }_2,\mathrm{\cdots },{\sigma }_k$ of $[n]$ satisfying, for all $i\in [k]$,

1. ${\sigma }_i$ has at least one node, and nodes in ${\sigma }_i$ have no connection between each other,

2. for any node in ${\sigma }_i$, it can be inhibited and excited only by nodes in ${\cup }_{j=1}^{i-1}{\sigma }_j$.

Thus, the nodes of ${\sigma }_1$ get neither inhibition nor excitation from other nodes, which means their dynamics is independent from the rest of the network. Hence, they all converge exponentially to their unique fixed point, $x_i^{*}=b_i$ for all $i\in {\sigma }_1$. Then, once nodes of ${\sigma }_1$ reached their equilibrium, nodes of ${\sigma }_2$ will received constant inputs. Hence, nodes of ${\sigma }_2$ will be stabilized by ${\sigma }_1$, which means ${\sigma }_1\cup {\sigma }_2$ has a unique globally asymptotically stable fixed point. So on and so forth and thus ${\cup }_{i=1}^k{\sigma }_i=[n]$ has a globally asymptotically stable fixed point.

The proof of Theorem 2 follows these lines. We study the fixed point and their stability first when their support is $\sigma =[n]$ and then when $\sigma ⊊[n]$ before concluding. However, we first need the following Lemma

We can consider a node $p\in \overline{\sigma }$ such that $p-1\in \sigma$. Thus ${\displaystyle x_{p-1}^{\ast }>0}$ and $x_p^{*}=0$. If the node $p-1$ was excitatory, then, as $x^{*}$ is solution of Equation 13, we would also have ${\displaystyle x_p^{\ast }>0}$. Therefore, node $p-1$ has to be inhibitory and thus from Lemma 1, we have for all ${\displaystyle t\ge 0}$,

Hence, according to the condition stated in Equation 4, ${\displaystyle x_p^{\ast }(t)>0}$ which contradicts the initial assumption $x_p^{*}=0$. Thus, under condition from Equation 4, the fixed point with $[n]$ is the only possible support.

We can consider a node $p\in \sigma$ such that $p-1\in \overline{\sigma }$. For the sake of clarity, as we consider cycles, we can say that $p=1$. Thus $x_n^{*}=0$ and ${\displaystyle x_1^{\ast }>0}$. If the node $n$ was excitatory, then, as $x^{*}$ is solution of Equation 13, we would also have $x_1^{*}=0$. Therefore, node $n$ has to be inhibitory and thus $x_1^{*}=b_1$.

Then we consider the path from node 1 to node $n$. By definition, we know that

Thus, we now compute the possible fixed point such that $x_1^{*}=b_1$.

Let us first consider the case in which $n$ is the only inhibitory node. Then, nodes $\{1,\mathrm{\dots },n-1\}$ are all excitatory. Hence, starting from $x_1^{*}=b_1$ and using the first line of Equation 16, we obtain that ${\displaystyle x_n^{\ast }=b_1\prod _{i=1}^{n-1}{w}_i>0}$, which contradicts $x_n^{*}=0$. Therefore, there is no possible fixed point on $\sigma ⊊[n]$ when $n_I=1$ under Equation 5.

Now, assume there are strictly more than one inhibitory node and an odd number. Therefore, $a_2-1$ is the next inhibitory node after node $a_1-1=0$ which if node $n$ when following the cycle. Then we have

which is null under condition from Equation 5. From the first line of Equation 16, all the following excited nodes will have null activity until the next inhibited node. Hence, $x_{a_2}^{*}=\mathrm{\cdots }=x_{a_3-1}^{*}=0$ and $x_{a_3}^{*}=b_{a_3}$ which is then in the same case as node a₁. We can thus compute by strong recurrence the possible fixed points. We check their existence by ensuring that the initial condition $x_n^{*}=0$ is still satisfied. From the recursive computation, we clearly see that if $x_{a_k-1}^{*}=0$, then ${\displaystyle x_{a_{k+1}-1}^{\ast }=b_{a_k}\prod _{i=a_k}^{a_{k+1}-1}{w}_i>0}$ (we used the convention given by Equation 2) and $x_{a_{k+2}-1}^{*}=0$. Therefore, the initial condition $x_n^{*}=0$ is only satisfied when the number of inhibitory nodes is even. Hence, the system with odd inhibitory nodes has no fixed point supported by $\sigma ⊊[n]$ under condition given by Equation 5.

When the number of inhibitory nodes is even, as the recurrence is on two successive inhibited nodes, there are two fixed points depending on the two possible initial conditions: ${\displaystyle x_{a_1}^{\ast }>0}$ or ${\displaystyle x_{a_2}^{\ast }>0}$. We denote by $x^{*,1}$ and $x^{*,2}$ these two fixed point having respectively ${\sigma }_1,{\sigma }_2⊊[n]$ as support where

In particular, ${\sigma }_1\cup {\sigma }_2=[n]$ and ${\sigma }_1\cap {\sigma }_2=\mathrm{\varnothing }$. Using the convention that ${\prod }_i^j\mathrm{\cdots }=1$ when ${\displaystyle j<i}$ and defining the function ${\displaystyle \varphi :[n]\to \mathcal{A}}$ such that ${\displaystyle \varphi (k)\in \mathcal{A}}$ is the last (following the cycle) inhibited node before $k$ (possibly $k$), we can give the exact formula of $x^{*,i}$, $i\in \{1,2\}$,

To check the stability, we again use linear theory. To do so, we still have to check the condition given by Equation 15 (only on inhibited nodes). From the computation of the fixed points, we have shown that under condition from Equation 5, for all ${\displaystyle a\in \mathcal{A}}$, either $x_a^{*}=0$ because ${\displaystyle b_a<w_{a-1}{x}_{a-1}^{\ast }}$ or ${\displaystyle x_a^{\ast }>0}$ because ${\displaystyle b_a>w_{a-1}{x}_{a-1}^{\ast }}$. Thus condition given by Equation 15 is satisfied. Hence, we now compute the eigenvalues of $-I_{{\sigma }_i}+W_{{\sigma }_i}$ for $i\in \{1,2\}$, which are the solutions of

To do so we write the weight matrix $W_{{\sigma }_i}$.

Both $W_{{\sigma }_1}$ and $W_{{\sigma }_2}$ are similar so we only write

We deduce that

so $\lambda =-1$ and both fixed points are asymptotically stable.

Eventually, when Equation 4 is satisfied, the system with even or odd inhibitory nodes has only one globally asymptotically stable fixed point on $[n]$. When $n_I$ is even and Equation 5 is satisfied, the system has only two asymptotically stable fixed points on $\sigma ⊊[n]$ and one unstable fixed point on $[n]$. When $n_I$ is odd, the system has an asymptotically stable fixed point on $[n]$ under conditions Equations 5 and 6; otherwise, it has a unique unstable fixed point on $[n]$ (thus no stable fixed point) under conditions Equations 5 and 7.