Network structure and input integration in competing firing rate models for decision-making

Abstract

Making a decision among numerous alternatives is a pervasive and central undertaking encountered by mammals in natural settings. While decision making for two-option tasks has been studied extensively both experimentally and theoretically, characterizing decision making in the face of a large set of alternatives remains challenging. We explore this issue by formulating a scalable mechanistic network model for decision making and analyzing the dynamics evoked given various potential network structures. In the case of a fully-connected network, we provide an analytical characterization of the model fixed points and their stability with respect to winner-take-all behavior for fair tasks. We compare several means of input integration, demonstrating a more gradual sigmoidal transfer function is likely evolutionarily advantageous relative to binary gain commonly utilized in engineered systems. We show via asymptotic analysis and numerical simulation that sigmoidal transfer functions with smaller steepness yield faster response times but depreciation in accuracy. However, in the presence of noise or degradation of connections, a sigmoidal transfer function garners significantly more robust and accurate decision-making dynamics. For fair tasks and sigmoidal gain, our model network also exhibits a stable parameter regime that produces high accuracy and persists across tasks with diverse numbers of alternatives and difficulties, satisfying physiological energetic constraints. In the case of more sparse and structured network topologies, including random, regular, and small-world connectivity, we show the high-accuracy parameter regime persists for biologically realistic connection densities. Our work shows how neural system architecture is potentially optimal in making economic, reliable, and advantageous decisions across tasks.

Appendices

Appendix A: Uniqueness of Fixed Point with Sigmoidal Gain

In this section of the Appendix, we provide a justification for the sufficient conditions guaranteeing a unique fixed point in the all-to-all network with sigmoidal gain discussed in Section 3.1.1, which we restate below.

Uniqueness of Fixed Point with Sigmoidal Gain:

The decision-making network model given in Eq. (1) with sigmoidal gain function f as prescribed by Eq. (2) has a unique fixed point if there exists positive constant M such that f^′(x_i) ≤ M,∀x_i, and $\frac {wM}{N-1} < 1$ for i = 1,…,N.We consider fixed points $x^{*} = (x_{1}^{*}, x_{2}^{*}, \dots , x_{N}^{*})$ and $x^{\prime } = (x_{1}^{\prime }, x_{2}^{\prime } , \dots , x_{N}^{\prime })$ for Eq. (1), and demonstrate that x^∗ = x^′ if $\frac {wM}{N-1} < 1$. Let $u_{i} = x_{i}^{\prime } - x_{i}^{*}$ denote the difference between the i th components of the fixed points. Since x^′ is a fixed point, it follows that $x^{\prime }_{i} = f(-{\sum }_{j\ne i} \frac {w}{N-1}x_{j}^{\prime } + S_{i})$. Hence,

$$ x_{i}^{*} + u_{i} = f\left( -\sum\limits_{j\ne i } \frac{w}{N-1}x_{j}^{*} + S_{i} - \sum\limits_{j\ne i} \frac{w}{N-1}u_{j}\right). $$

(14)

Since f is smooth, according to the mean value theorem

$$\begin{array}{@{}rcl@{}} &&f\left( -\sum\limits_{j\ne i } \frac{w}{N-1}x_{j}^{*} + S_{i} - \sum\limits_{j\ne i} \frac{w}{N-1}u_{j}\right) \\ &=& f\left( - \sum\limits_{j\ne i } \frac{w}{N-1}x_{j}^{*} + S_{i}\right) + f^{\prime}(c_{i})\left( - \sum\limits_{j\ne i} \frac{w}{N-1}u_{j}\right), \end{array} $$

where c_i is contained in open interval $(-{\sum }_{j\ne i } \frac {w}{N-1}x_{j}^{*} + S_{i} - {\sum }_{j\ne i} \frac {w}{N-1}u_{j}, -{\sum }_{j\ne i } \frac {w}{N-1}x_{j}^{*} + S_{i})$. Given $x_{i}^{*}$ is a fixed point, we re-express Eq. (14) as

$$\begin{array}{@{}rcl@{}} x_{i}^{*} + u_{i} &=& \!f\left( - \sum\limits_{j\ne i } \frac{w x_{j}^{*}}{N - 1} + S_{i}\right) + f^{\prime}(c_{i})\left( - \sum\limits_{j\ne i} \frac{w u_{j}}{N - 1}\right) \\ \implies u_{i} &=& -\frac{w}{N-1}f^{\prime}(c_{i})\left( \sum\limits_{j\ne i}u_{j}\right). \end{array} $$

Labeling $a_{i} = \frac {w}{N-1}f^{\prime }(c_{i})$ for notational convenience, it follows $u_{i} + a_{i}{\sum }_{j\ne i}u_{j} = 0$. We can rewrite this compactly in matrix notation as Au = 0, where

$$A = \left[\begin{array}{lllll} 1 & a_{1} & a_{1} & {\dots} & a_{1} \\ a_{2} & 1 & a_{2} & {\dots} & a_{2} \\ a_{3} & a_{3} & 1 & {\dots} & a_{3} \\ {\vdots} & {\vdots} & {\vdots} & {\ddots} & {\vdots} \\ a_{N} & a_{N} & a_{N} & {\dots} & 1 \end{array}\right], $$

and $\det (A) = {\prod }_{k = 1}^{N} (1-a_{k}) + {\sum }_{i = 1}^{N} a_{i} {\prod }_{k\ne i} (1-a_{k})$ (Horn and Johnson 2012).

Finally, assuming f^′(x_i) ≤ M,∀x_i, and $\frac {w}{N-1}M < 1$, it follows that $a_{i} = \frac {w}{N-1}f^{\prime }(c_{i}) \leq \frac {w}{N-1}M < 1$. As a result, det(A)≠ 0, and the linear system Au = 0 has a unique solution u = 0. This implies that x^′ = x^∗, demonstrating the fixed point is indeed unique.

Appendix B: Global Attraction to the Perturbed Fixed Point with Binary Gain

In Appendix B, we provide a justification for the conditions guaranteeing the existence and global attraction of the perturbed fixed point in the all-to-all network with binary gain discussed in Section 3.1.2, which we restate below.

Fig. 15

Impact of noise on accuracy for networks with sigmoidal gain and hard difficulty tasks. Numerically computed accuracy dependence on number of alternatives, N, and lateral inhibition strength, w, for a fully-connected network with sigmoidal gain and noise of strength: aσ = 2^− 8, bσ = 2^− 7, cσ = 2^− 6, dσ = 2^− 5, eσ = 2^− 4, fσ = 2^− 3, gσ = 2^− 2, and hσ = 2^− 1. Each plot depicts the accuracy averaged over 100 fair initial conditions and tasks of hard difficulty

Global Attraction to the Perturbed Fixed Point with Binary Gain:

For non-simple and fair tasks with number of alternatives N ≥ 3, the reduced decision-making network model given in Eq. (5) with binary gain function f as prescribed by Eq. (3) is globally attracted to the perturbed fixed point such that

$$ \displaystyle \lim_{t\to \infty} x_{w} = 1, $$

(6a)

$$ \displaystyle \lim_{t\to \infty} x_{l} \in [z,z+\epsilon), $$

(6b)

for any 𝜖 > 0 under assumptions

$$ N > \max \left( \frac{2w-(S_{l}-b)}{w-(S_{l}-b)}, \frac{w}{S_{l}-b}+ 1\right), $$

(7a)

$$ w > S_{l}-b, $$

(7b)

where

$$ z = \frac{(N-1)(S_{l}-b)-w}{(N-2)w}. $$

(8)

Under assumptions (7), we first show that it is impossible for $\frac {dx_{l}}{dt}= 0$ and thus impossible for stationary dynamics to be reached. If $\frac {dx_{l}}{dt} = 0$, then either (i) x_l = 0 and $f\left (-wx_{l} - \frac {w}{N-1} (x_{w}-x_{l}) + S_{l}\right )= 0$, or (ii) x_l = 1 and $f\left (-wx_{l} - \frac {w}{N-1} (x_{w}-x_{l}) + S_{l}\right ) = 1$. We verify that neither of these cases are possible. Case (i) requires $x_{w} > \frac {(S_{l}-b)(N-1)}{w}$, but assumption (7a) requires w < (S_l − b)(N − 1), forcing x_w > 1 and making this potential fixed point impossible. Case (ii) requires $-wx_{l} - \frac {w}{N-1} (x_{w}-x_{l}) + S_{l} \geq b$, making $x_{w} \leq \frac {(S_{l}-b)(N-1)}{w} + 1 - (N-1)$. However, assumption (7b) requires that $(N-1) > \frac {(S_{l}-b)(N-1)}{w} + 1$, forcing x_w < 0, which is impossible.

Next, to determine the state around which the system gravitates after a long time horizon, we show $z_{l} = \frac {(N-1)(S_{l}-b) - wx_{w}}{(N-2)w}$ separates the dynamics of x_l into distinct strictly increasing and strictly decreasing regimes, such that if x_l ≤ z_l then $\frac {dx_{l}}{dt} >0$ and if x_l > z_l then $\frac {dx_{l}}{dt}<0$. To see this, note that if $\frac {dx_{l}}{dt}>0$, then $f\left (-wx_{l} - \frac {w}{N-1} (x_{w}-x_{l}) + S_{l}\right ) = 1$ necessarily, and thus $-wx_{l} - \frac {w}{N-1} (x_{w}-x_{l}) + S_{l} \geq b$. This forces $x_{l} \leq \frac {(N-1)(S_{l}-b) - wx_{w}}{(N-2)w}$, which demonstrates $x_{l} > z_{l} \implies \frac {dx_{l}}{dt} < 0$. An analogous argument assuming instead $\frac {dx_{l}}{dt}<0$ yields $x_{l} \leq z_{l} \implies \frac {dx_{l}}{dt} > 0$.

As a result, it is guaranteed after sufficient time elapses x_l → [z_l,z_l + 𝜖), for any 𝜖 > 0. Since z_l ∈ (0, 1) under assumptions (7) and x_w ∈ [0, 1], we obtain, independent of x_w, the upper bound $z_{l} \leq \frac {(N-1)(S_{l}-b)}{(N-2)w} < \frac {S_{l}-b}{w}$. Thus, as t →∞, the total input into the gain function for the w th node is − wx_l + S_w > b + (S_w − S_l) > b, and consequently x_w → 1. As a result, x_l → [z,z + 𝜖) for $z = \frac {(N-1)(S_{l}-b)-w}{(N-2)w}$, which gives an attracting perturbed fixed point depending only on the model parameters.

Appendix C: Robustness of Decision-Making for Hard Difficulty Tasks

Fig. 16

Impact of noise on accuracy for networks with binary gain and hard difficulty tasks. Numerically computed accuracy dependence on number of alternatives, N, and lateral inhibition strength, w, for a fully-connected network with binary gain and noise of strength: aσ = 2^− 8, bσ = 2^− 7, cσ = 2^− 6, dσ = 2^− 5, eσ = 2^− 4, fσ = 2^− 3, gσ = 2^− 2, and hσ = 2^− 1. Each plot depicts the accuracy averaged over 100 fair initial conditions and tasks of hard difficulty

Fig. 17

Impact of connection removal on accuracy for networks with sigmoidal gain and hard difficulty tasks. Numerically computed accuracy dependence on number of alternatives, N, and lateral inhibition strength, w, for a fully-connected network with sigmoidal gain and connection removal of probability: a 0.003, b 0.01, c 0.1, d 0.2, e 0.4, and f 0.8. Each plot depicts the accuracy averaged over 100 fair initial conditions and tasks of hard difficulty

Fig. 18

Impact of connection removal on accuracy for networks with binary gain and hard difficulty tasks. Numerically computed accuracy dependence on number of alternatives, N, and lateral inhibition strength, w, for a fully-connected network with binary gain and connection removal of probability: a 0.003, b 0.01, c 0.1, d 0.2, e 0.4, and f 0.8. Each plot depicts the accuracy averaged over 100 fair initial conditions and tasks of hard difficulty

In this Appendix, we provide additional figures depicting the network model decision-making dynamics subject to noise and then connection removal for hard difficulty tasks. In Figs. 15 and 16, we plot, for various choices of noise strength, the accuracy over the w − N parameter space for all-to-all networks with sigmoidal and binary gain functions, respectively. Similarly, in Figs. 17 and 18, we compare the decision-making performance upon randomly removing connections with increasing probability. The resultant dynamics are analogous to those evoked in the medium difficulty case discussed in detail in Section 3.4.