Local excitation solutions in one-dimensional neural fields by external input stimuli

Abstract

Cortical neurons are massively connected with other cortical and subcortical cells, and they receive synaptic inputs from multiple sources. To explore the basis of how interconnected cortical cells are locally activated by such inputs, we theoretically analyze the local excitation patterns elicited by external input stimuli by using a one-dimensional neural field model. We examine the conditions for the existence and stability of the local excitation solutions under arbitrary time-invariant inputs and establish a graphic analysis method that can detect all steady local excitation solutions and examine their stability. We apply this method to a case where a pair of supra- and subthreshold stimuli are applied to nearby positions in the field. The results demonstrate that there can exist bistable local excitation solutions with different lengths and that the local excitation exhibits hysteretic behavior when the relative distance between the two stimuli is altered.

Appendices

Appendix A: Proof of Theorem 2

• (1) When S ^*_x1  > S ^*_x2 and $w(a^{*} )\left( {S_{x1}^{*} - S_{x2}^{*} } \right) + S_{x1}^{*} S_{x2}^{*} < 0$ hold, we find C > 0 in (10) by using u ^*_x1  > 0 and u ^*_x2  < 0. Differentiating (3) with respect to x yields

  $$ \frac{{{\text{d}}\bar{u}(x)}}{{{\text{d}}x}} = w(x - x_{1}^{*} ) - w(x - x_{2}^{*} ) + \frac{{{\text{d}}S(x)}}{{{\text{d}}x}}. $$

  (A.1)

  Then, by substituting x = x ^*₁ and x ^*₂ , we obtain

  $$ u_{x1}^{*} = w(0) - w(a^{*} ) + S_{x1}^{*} , $$

  (A.2)
  $$u_{x2}^{*} = - w(0) + w(a^{*} ) + S_{x2}^{*} . $$

  (A.3)

  From these equations, (9) can be transformed into

  $$ \begin{aligned} \begin{array}{*{20}c} B \\ \end{array} &= \frac{1}{{\tau^{{}} u_{x1}^{*} u_{x2}^{*} }}\left[ {w(0)\left\{ {2w(a^{*} ) - S_{x1}^{*} + S_{x2}^{*} } \right\} - 2w(a^{*} )^{2} + 2\left\{ {(S_{x1}^{*} - S_{x2}^{*} )w(a^{*} ) + S_{x1}^{*} S_{x2}^{*} } \right\}} \right] \\ &= \frac{1}{{\tau^{{}} u_{x1}^{*} u_{x2}^{*} }}\left[ {w(0)\left\{ {2w(a^{*} ) - S_{x1}^{*} + S_{x2}^{*} } \right\} - 2w(a^{*} )^{2} } \right] + 2\tau C. \end{aligned}$$

  (A.4)

  Hence, by using the relation

  $$ 2w(a^{*} ) - S_{x1}^{*} + S_{x2}^{*} < - \frac{{2S_{x1}^{*} S_{x2}^{*} }}{{S_{x1}^{*} - S_{x2}^{*} }} - S_{x1}^{*} + S_{x2}^{*} = - \frac{{S_{x1}^{*2} + S_{x2}^{*2} }}{{S_{x1}^{*} - S_{x2}^{*2} }} < 0 $$

  (A.5)

  and the property of the connectivity w(0) > 0, we can find B > 0. Therefore, both the coefficients B and C of the characteristic function are positive, so that the differential equation (8) is stable.

• (2) When $w(a^{*} )\left( {S_{x1}^{*} - S_{x2}^{*} } \right) + S_{x1}^{*} S_{x2}^{*} > 0$ holds, we find C < 0 from (10) by using u ^*_x1  > 0 and u ^*_x2  < 0. Thus, the system is unstable.

  Furthermore, when S ^*_x1  > S ^*_x2 and $w(a^{*} )\left( {S_{x1}^{*} - S_{x2}^{*} } \right) + S_{x1}^{*} S_{x2}^{*} \le 0$ hold, we find

  $$ w(a^{*} ) \ge - \frac{{S_{x1}^{*} S_{x2}^{*} }}{{S_{x1}^{*} - S_{x2}^{*} }}. $$

  (A.6)

Hence,

$$ w(a^{*} ) - S_{x1}^{*} \ge - \frac{{S_{x1}^{*} S_{x2}^{*} }}{{S_{x1}^{*} - S_{x2}^{*} }} - S_{x1}^{*} = - \frac{{S_{x1}^{*2} }}{{S_{x1}^{*} - S_{x2}^{*} }} \ge 0, $$

(A.7)

$$ w(a^{*} ) + S_{x2}^{*} \ge - \frac{{S_{x1}^{*} S_{x2}^{*} }}{{S_{x1}^{*} - S_{x2}^{*} }} + S_{x2}^{*} = - \frac{{S_{x2}^{*2} }}{{S_{x1}^{*} - S_{x2}^{*} }} \ge 0. $$

(A.8)

Note that at least either S ^*_x1 or S ^*_x2 must take a value different from 0 because of S ^*_x1  < S ^*_x2 . Therefore, we obtain B < 0 from (9), so that the system is unstable.

Appendix B: The method for plotting the $a - \hat{S}$ curve

In this appendix, we show the method for plotting the $a - \hat{S}$ curve from the function S(x) according to Definition 1. First, we give some definitions.

Definition 2

We say that f(x) is a monotone increasing (decreasing) function if f(x₁) < f(x₂) (f(x₁) > f(x₂) ) for any x₁, x₂with x₁ < x₂in its domain. We refer to a monotone increasing or decreasing function as a monotone function. We also say that f(x) is a constant function if f(x₁) = f(x₂) for any x₁and x₂in its domain.

Definition 3

We define S _i (x) (i = 1,…, N) to be functions that satisfy the following three conditions, where a finite interval [d _i , d_i+1] is the domain of S _i (x). We refer to the function S _i (x) as a subfunction of S(x).

• (1) S _i (x) = S(x) for all i,

• (2) S _i (x) is either a monotone or constant function,

• (3) The domain of the neural field [x_min, x_max] is covered by the domain of subfunctions, i.e., d₁ = x_minand d_N+1=x_max.

Definition 3 means the division of S(x) into N subfunctions S _i (x) (i = 1,…, N) such that each subfunction is either a monotone or constant function. S₁(x) − S₇(x) in Fig. 6a shows an example of the subfunctions for N = 7 corresponding to S(x) in Fig. 3a.

Fig. 6

The way of plotting the $a - \hat{S}$ curve. a According to Definition 3, the subfunctions S ₁(x) − S ₇(x) are constructed from function S(x) in Fig. 3a such that each subfunction is either a monotone or constant function. b The $a - \hat{S}$ curve has been drawn by plotting Γ _ij (i = 1,…, 7, j = 1,…, 7) by using Theorem 5 from the subfunctions S ₁(x) − S ₇(x). The line of $\hat{S}$ = 0 contains Γ _ij for many pairs of i, j, but their labels are omitted

Definition 4

We define Γ _ij (i = 1,…, N, j = 1,…, N) to be a set of ($a,\hat{S}$) such that there exist x₁and x₂satisfying the following relations:

• $$ (1) \quad x_{1} \in D_{i} ,\quad x_{2} \in D_{j} , $$

  (B.1)
• $$ (2) \quad x_{1} < x_{2} , $$

  (B.2)
• $$ (3) \quad S_{i} (x_{1} ) = S_{j} (x_{2} ) = \hat{S}, $$

  (B.3)
• $$ (4) \quad a = x_{2} - x_{1} , $$

  (B.4)

  where D _i ≡[d _i , d_i+1] is the domain of the subfunction S(x).

From Definitions 1 and 4, we can find that Γ _ij is a subset of the $a - \hat{S}$ curve and that the $a - \hat{S}$ curve is described as $\bigcup\nolimits_{i,j} {\Upgamma_{ij} } .$

Definition 5

Let S ⁻¹ _Li (S) and S ⁻¹ _Hi (S) be functions such that

$$ S_{Li}^{ - 1} (S) = \left\{ \begin{gathered} S_{i}^{ - 1} (S),\quad{\text{if}}\,S_{i} \,{\text{is}}\,{\text{a}}\,{\text{monotone}}\,{\text{function,}} \hfill \\ d_{i} ,\quad{\text{if}}\,S_{i} \,{\text{is}}\,{\text{a constant}}\,{\text{function}}, \hfill \\ \end{gathered} \right. $$

(B.5)

$$ S_{Hi}^{ - 1} (S) = \left\{ \begin{gathered} S_{i}^{ - 1} (S),\quad{\text{if}}\,S_{i} \,{\text{is}}\,{\text{a}}\,{\text{monotone}}\,{\text{function,}} \hfill \\ d_{i + 1} ,\quad{\text{if}}\,S_{i} \,{\text{is}}\,{\text{a constant}}\,{\text{function}}, \hfill \\ \end{gathered} \right. $$

(B.6)

where S ⁻¹ _i (S) denotes an inverse function of S _i (x).

From the above definitions, we have the following theorem that gives the explicit description of Γ _ij .

Theorem 5

Let R _i be the range of a subfunction S _i (x). Then, Γ _ij is described as follows:

• (1) When both S _i and S _j (i < j) are monotone functions and R _i  ∩ R_j≠ϕ,

  $$ \Upgamma_{ij} = \left\{ {(a,\hat{S})|a = S_{j}^{ - 1} (\hat{S}) - S_{i}^{ - 1} (\hat{S}),\,a > 0,\,\hat{S} \in R_{i} \cap R_{j} } \right\}, $$

  (B.7)

• (2) When S _i and/or S _j (i ≤ j) are constant functions and R _i  ∩ R_j≠ϕ,

  $$ \Upgamma_{ij} = \left\{ {(a,\hat{S})|a \in [S_{Lj}^{ - 1} (S_{c} ) - S_{Hi}^{ - 1} (S_{c} ),\,S_{Hj}^{ - 1} (S_{c} ) - S_{Li}^{ - 1} (S_{c} )]\begin{array}{*{20}c} , \\ \end{array} \quad a > 0,\,\hat{S} = S_{c} } \right\}, $$

  (B.8)

  where S _c is defined such that {S _c } = R _i  ∩ R _j ,

• (3) Otherwise, Γ _ij  = ϕ.

Proof of Theorem 5

We can prove Γ _ij  = ϕ from Definition 4 in case of (1) i > j, (2) R _i  ∩ R _j≠ϕ, and (3) S _i is a monotone function and i = j. Thus, by excluding these cases, we consider the following five cases,

• Case A: Both S _i and S _j are monotone functions, i < j, and R _i  ∩ R _j≠ϕ,

• Case B: S _i is a monotone function, S _j is a constant function, i < j, and R _i  ∩ R _j≠ϕ,

• Case C: S _i is a constant function, S _j is a monotone function, i < j, and R _i  ∩ R _j≠ϕ,

• Case D: S _i is a constant function and i = j,

• Case E: Both S _i and S _j are constant functions, i < j, and R _i  ∩ R _j≠ϕ.

Then, for each case, Γ _ij is given by the following lemma. (Proof of the lemma is given later.)

Lemma 1

Let S _ci be the value of a subfunctionS _i (x) when S _i (x) is a constant function. Then, for each case, Γ _ij is described as follows:

$$ \Upgamma_{ij} = \left\{ {(a,\hat{S})|a = S_{j}^{ - 1} (\hat{S}) - S_{i}^{ - 1} (\hat{S}),\,a > 0,\,\hat{S} \in R_{i} \cap R_{j} } \right\}\quad {\text{for}}\;{\text{Case}}\;{\text{A}}, $$

(B.9)

$$ \Upgamma_{ij} = \left\{ {(a,\hat{S})|a \in [d_{j} - S_{i}^{ - 1} (S_{cj} ),\,d_{j + 1} - S_{i}^{ - 1} (S_{cj} )],\,a > 0,\,\hat{S} = S_{cj} } \right\}\quad {\text{for}}\;{\text{Case}}\;{\text{B}}, $$

(B.10)

$$ \Upgamma_{ij} = \left\{ {(a,\hat{S})|a \in [S_{j}^{ - 1} (S_{ci} ) - d_{i + 1} ,\,S_{j}^{ - 1} (S_{ci} ) - d_{i} ],\,a > 0,\,\hat{S} = S_{ci} } \right\}\quad {\text{for}}\;{\text{Case}}\;{\text{C}}, $$

(B.11)

$$ \Upgamma_{ij} = \left\{ {(a,\hat{S})|a \in (0,d_{i + 1} - d_{i} ],\,\hat{S} = S_{ci} } \right\}\quad {\text{for}}\;{\text{Case}}\;{\text{D}}, $$

(B.12)

$$ \Upgamma_{ij} = \left\{ {(a,\hat{S})|a \in [d_{j} - d_{i + 1} ,\,d_{j + 1} - d_{i} ],\,a > 0,\,\hat{S} = S_{ci} } \right\}\quad {\text{for}}\;{\text{Case}}\;{\text{E}}. $$

(B.13)

From Lemma 1, we can see that Γ _ij for Cases A is the same as (B.7), and that Γ _ij for Cases B-E are summarized as (B.8) by using the notations of S ⁻¹ _Li and S ⁻¹ _Hi in Definition 5, so that we obtain Theorem 5. □

The proof of Lemma 1 is given as follows.

Proof of Lemma 1

Since proofs of Cases A − E are similar, we show only proof of Case A and omit proofs of the other cases.

If $(a,\hat{S})$ $\in \Upgamma_{ij}$, there exist x ₁ and x ₂ that satisfy (B.1)–(B.4) from the definition of Γ _ij . We can find $\hat{S} \in R_{i} \cap R_{j}$ from (B.3) and $a = S_{j}^{ - 1} (\hat{S}) - S_{i}^{ - 1} (\hat{S})$ from (B.3) and (B.4). a > 0 also holds from (B.2) and (B.4), so that

$$ (a,\hat{S}) \in \left\{ {(a,\hat{S})|a = S_{j}^{ - 1} (\hat{S}) - S_{i}^{ - 1} (\hat{S}),\,a > 0,\,\hat{S} \in R_{i} \cap R_{j} } \right\}. $$

(B.14)

On the contrary, if (B.14) holds, we can obtain (B.1)–(B.4) by setting $x_{1} = S_{i}^{ - 1} (\hat{S})$ and $x_{2} = S_{j}^{ - 1} (\hat{S})$. Thus, we have $(a,\hat{S}) \in \Upgamma_{ij}$. □

Since the $a - \hat{S}$ curve is $\bigcup\nolimits_{i,j} {\Upgamma_{ij} }$ as stated above, we can draw the $a - \hat{S}$ curve by plotting points $(a,\hat{S}) \in \Upgamma_{ij}$ for every i, j with Γ _ij ≠ϕ by using Theorem 5. Figure 6b shows how the $a - \hat{S}$ curve is composed of Γ _ij , where each curve Γ _ij has been obtained from S _i (x) and S _j (x) depicted in Fig. 6a.

Appendix C: The method for finding solutions of the steady condition 1

Here we show how the solutions of the steady condition 1 can be obtained when the intersections of the $a - \hat{S}$ curve with Y(a) are given. All the definitions in Appendix B (Definitions 2–5) are also used in this appendix.

As mentioned in Appendix B, the $a - \hat{S}$ curve is written as $\bigcup\nolimits_{i,j} {\Upgamma_{ij} }$ by using Γ _ij in Definition 4. Therefore, when a point $(a,\hat{S})$ lies on the $a - \hat{S}$ curve, $(a,\hat{S}) \in \Upgamma_{ij}$ holds for some pair of i, j, and there exist x ₁ and x ₂ satisfying (B.1)–(B.4). Hence, we denote a set of these variables (x ₁, x ₂) by $\Uptheta_{ij} [a,\hat{S}]$. The following theorem shows the explicit description of $\Uptheta_{ij} [a,\hat{S}]$.

Theorem 6

$\Uptheta_{ij} [a,\hat{S}]$ is given as follows:

• (1) When S _i and/or S _j are monotone functions,

  $$ \Uptheta_{ij} [a,\hat{S}] = \left\{ {(x_{1} ,x_{2} )|x_{1} = \max \left( {S_{Li}^{ - 1} (\hat{S}),\,S_{Lj}^{ - 1} (\hat{S}) - a} \right),\,x_{2} = x_{1} + a} \right\}, $$

  (C.1)

• (2) When both S _i and S _j are constant functions,

  $$ \Uptheta_{ij} [a,\hat{S}] = \left\{ {(x_{1} ,x_{2} )|x_{1} \in \left[ {\max \left( {S_{Li}^{ - 1} (\hat{S}),\,S_{Lj}^{ - 1} (\hat{S}) - a} \right),\,\min \left( {S_{Hi}^{ - 1} (\hat{S}),\,S_{Hj}^{ - 1} (\hat{S}) - a} \right)} \right],\,x_{2} = x_{1} + a} \right\}, $$

  (C.2)

  where S _i and S _j are the subfunctions defined in Definition 3. S ⁻¹ _Li and S ⁻¹ _Hi are the functions defined in Definition 5.

Proof of Theorem 6

We present the following lemma. (The proof of this lemma is shown later.)

Lemma 2

Consider the same classification as Cases A–E shown in proof of Theorem 5 in Appendix B , then$\Uptheta_{ij} [a,\hat{S}]$for each case is given as follows:

$$ \Uptheta_{ij} [a,\hat{S}] = \left\{ {(x_{1} ,x_{2} )|x_{1} = S_{i}^{ - 1} (\hat{S}),\,x_{2} = x_{1} + a} \right\}\quad {\text{for}}\;{\text{Case}}\;{\text{A}}, $$

(C.3)

$$ \Uptheta_{ij} [a,\hat{S}] = \left\{ {(x_{1} ,x_{2} )|x_{1} = S_{i}^{ - 1} (\hat{S}),\,x_{2} = x_{1} + a} \right\}\quad {\text{for}}\;{\text{Case}}\;{\text{B}}, $$

(C.4)

$$ \Uptheta_{ij} [a,\hat{S}] = \left\{ {(x_{1} ,x_{2} )|x_{1} = S_{j}^{ - 1} (\hat{S}) - a,\,x_{2} = x_{1} + a} \right\}\quad {\text{for}}\;{\text{Case}}\;{\text{C}}, $$

(C.5)

$$ \Uptheta_{ij} [a,\hat{S}] = \left\{ {(x_{1} ,x_{2} )|x_{1} \in [d_{i} ,d_{i + 1} - a],\,x_{2} = x_{1} + a} \right\}\quad {\text{for}}\;{\text{Case}}\;{\text{D}}, $$

(C.6)

$$ \Uptheta_{ij} [a,\hat{S}] = \left\{ {(x_{1} ,x_{2} )|x_{1} \in [\max (d_{i} ,d_{j} - a),\min (d_{i + 1} ,\,d_{j + 1} - a)],\,x_{2} = x_{1} + a} \right\}\quad {\text{for}}\;{\text{Case}}\;{\text{E}}, $$

(C.7)

where [d _i , d_i+1] is the domain of the subfunction S _i (x) in Definition 3.

If we use the notations of S ⁻¹ _Li and S ⁻¹ _Hi in Definition 5, then Cases A–E in Lemma 2 can be summarized as Theorem 6. □

The proof of Lemma 2 is given as follows.

Proof of Lemma 2

Since proofs of Cases A–E are similar, we show only proof of Case A and omit the proofs of the other cases. From $(a,\hat{S}) \in \Upgamma_{ij}$ and Lemma 1, we have

$$ a = S_{j}^{ - 1} (\hat{S}) - S_{i}^{ - 1} (\hat{S}), $$

(C.8)

$$ a > 0, $$

(C.9)

$$ \hat{S} \in R_{i} \cap R_{j} . $$

(C.10)

If $(x_{1} ,x_{2} ) \in \Uptheta_{ij} [a,\hat{S}]$, we find $x_{1} = S_{i}^{ - 1} (\hat{S})$ and $x_{2} = S_{j}^{ - 1} (\hat{S})$ from (B.3). By using (C.8), we have x ₂ = x ₁ + a, so that

$$ (x_{1} ,x_{2} ) \in \left\{ {(x_{1} ,x_{2} )|x_{1} = S_{i}^{ - 1} (\hat{S}),\,x_{2} = x_{1} + a} \right\}. $$

(C.11)

On the contrary, if (C.11) holds, we can prove that (B.1)–(B.4) also hold by using (C.8) and (C.9). Thus, we obtain $(x_{1} ,x_{2} ) \in \Uptheta_{ij} [a,\hat{S}].$ □

Consider an intersection point (a ^* _k , S ^* _k ) of the $a - \hat{S}$ curve with Y(a) as in Step 1 of the graphic analysis method. Let us define i _k and j _k to be the integers satisfying (a ^* _k , S ^* _k ) ∈ $\Upgamma_{{i_{k} j_{k} }}$, and set $\Uptheta_{k} = \Uptheta_{{i_{k} j_{k} }} [a_{k}^{*} ,S_{k}^{*} ]$. Then, from the definition of $\Uptheta_{ij} [a,\hat{S}]$, we can find a ^* _k  = x ^*_2,k  − x ^*_1,k and S(x ^*_1,k ) = S(x ^*_2,k ) = S ^* _k for (x ^*_1,k , x ^*_2,k ) ∈ Θk. Since we can have the following corollary from Theorem 3, $\left( {x_{1}^{*} ,x_{2}^{*} } \right) \in \bigcup\nolimits_{k} {\Uptheta_{k} }$ are the solutions of the steady condition 1 in Theorem 1.

Corollary 1

The steady condition 1 holds for x ^*₁ , x ^*₂ if and only if

$$ (x_{1}^{*} ,x_{2}^{*} ) \in \bigcup\limits_{k} {\Uptheta_{k} } . $$

Proof of Corollary 1

If the steady condition 1 holds for x ^*₁ , x ^*₂ (x ^*₁  < x ^*₂ ), the point (a ^*, S^*) with a ^* = x ^*₂  − x ^*₁ and S ^* = S(x ^*₂ ) − S(x ^*₁ ) is an intersection of the $a - \hat{S}$ curve with Y(a) from Theorem 3.

We find $S_{i} (x_{1}^{*} ) = S_{j} (x_{2}^{*} ) = S^{*}$ with i, j satisfying x ^*₁  ∈ D _i and x ^*₂  ∈ D _j , where S _i and D _i (i = 1,…, N) denote the subfunction and its domain in Definition 3. Hence, (a ^*, S ^*) ∈ Γ _ij holds from Definition 4. Since a ^* = a ^* _k , S ^* = S ^* _k , i = i _k , and j = j _k hold for some k, we can obtain

$$ x_{1}^{*} \in D_{{i_{k} }} ,\quad x_{2}^{*} \in D_{{j_{k} }} , $$

(C.12)

$$ S_{{i_{k} }} (x_{1}^{*} ) = S_{{j_{k} }} (x_{2}^{*} ) = S_{k}^{*} , $$

(C.13)

$$ a_{k}^{*} = x_{2}^{*} - x_{1}^{*} . $$

(C.14)

Therefore, we can find $(x_{1}^{*} ,x_{2}^{*} ) \in \Uptheta_{k}$.

On the contrary, if $(x_{1}^{*} ,x_{2}^{*} ) \in \Uptheta_{k}$ holds for some k, (C.13) and (C.14) hold. Since $(a_{k}^{*} ,S_{k}^{*} )$ lies on an intersection of the $a - \hat{S}$ curve with Y(a), we obtain the steady condition 1 from Theorem 3. □

Since the elements of $\Uptheta_{k} = \Uptheta_{{i_{k} j_{k} }} [a_{k}^{*} ,S_{k}^{*} ]$ can be obtained by using Theorem 6 for each k, we can actually find the solutions of the steady condition 1. Note that, Theorem 6 indicates that, if both $S_{{i_{k} }}$ and $S_{{j_{k} }}$ are constant functions, Θ _k contains infinite number of elements. Thus, in this case, we need to pick up some elements $(x_{1,k}^{*} ,x_{2,k}^{*} ) \in \Uptheta_{k}$ by the required accuracy in order to plot $G[x;x_{1,k}^{*} ,x_{2,k}^{*} ]$ in Step 2 of the graphic analysis method.