Upload
vunhu
View
222
Download
0
Embed Size (px)
Citation preview
33
������������ ������ � � �� ��
�������! #"$����%&"$'(�)%���*$+,+.-/-0�21435��6/�5�27) �*.�,�� +,*$8)�9���2 ���:<;��=�$>=6?>@�A+B6/*C>D �-?��+B�A����7�1EGFIH<HKJ �� �L�M@�A�,L@75�5MANO'5-?���B �*P��6/QA+,6�;��B �-?��+A���)���.35��*C>@6��.Q,6?�2;+,�9 �-0��+,��RSM@�,6�+.6?M@8)+T"$��8)�F�UVJ ����+���-?� FXWYKJ �Z[75��6\ �*.�����A-?�21/:]��'[^_���D`/6�-/�D`�*$+,�a1bR F�cedKJ ��'f���)^?��;\�fgh:�8@���27)6/*C> FiHkjVJaEGFiHeWlJaEGFIHkUhJ�)*,:<7)�AZ,6�1?-4RG�� �L$M)�A�mL@75�5MANO'[-0���b �*9 �-0��+,�D�fgh:��)6?>,���=6/*C>D�AM@��+)75-k \�)+nQ$��*$��;��$��60>@�,+D6�*C>10^?+,�=��75�,*$+B3���1o��Q,+A�,+=6/*CMD�$>@��^?+p��>b1?��^?�m10^?+9��:��21qM!1#",����Q,�5>b1?^?+=��75�,*$+D �*.",�);< �8)�D�fgh:
R �-0��+,�r *s10^?+,�b��75�A*$+D6k;�LA+D�$>D",��75��Q�M@+D`/%�1_� <:&�A��6?>!75�� t>a b�A%/6�:�+� �*P �-?��+P�A��7�1=�fgh:&�!75���u�)��:<10����>Pv/��M@�) &3���Q$�2 �8a 4�a7w��*P`/6�-1q��*$��8! = ���-0'�",��1q��*$��8�6k;�LA+$�x1?6.35��6/%��5Z��)%/��6/^?+y��Z��)*$�5'zR F�Ue{|J �)6/Q) ����}��60��Q� �*~3���:�6?'[��`/+AL
R L,��L@10�b������^_M!7w�2;3���10��������6?��*$�5M@�A��",��7w��6?����7������,*$69 �*~������^_�21?�5M@���$�27w��%��,���@7�10�! =6��$�����,��75QA*,;D�)*$�5'
R�",��6�8A>,� F�UedKJ -k ��;<75���� �M EGFi��dKJ �5M@L@75�� k�2;�+$��L EGF�chJ �,��+@:� �*p",�,-0�21/:<1o>A��^?+@ �3��,��7��,�5>)�
�z���K�$�C�h���z�w�&���@���$�,���|���������4�l�h�z���C�|���4�h�A�,�V� �,�5�
�����C� �,�5�,�5�
�%D��>�35�21?*$8a1p>@��10�a b*$�C�A>,�2 �+)1=�a75��1?�a &��-/%k1q�,-?��>)+��@10%/60��+�-/-0�21?�.35��6/�5�275�.���2 ���:<;R F�Ue�KJ -/-0�21�35��6/�5�27_ �*B",����+A��%/�t",��%/�� ��)�,�
"$>�M)6�RG�a ��2:<;���� �>)��^/75Q$�2;=��6��� * "all-or-none" �P�)7w��%/�¡�275����",����6?>!75��10��",�� �-?��+) DLA��L@1?�RG�a ��2:<;�r �>)��^/75Q$�2;�6/^_����Z�>DM@6�",���5��L,+=v�L@ � <:�+D�a:��5'5+D35��6/�5�27� ��)L,��75%/�
�A6?��^/1#",���� �+,*$8D�����$��>DQ@ ���MD35��6/�5�275�BRGQ) <; Q) �MDQ@75+@ k>@%�35��6/�5�27��$>b6?>)�@ �3��,��7�L$'a3�;���>a1R�¢ H R U v���:�L��A>@6�£)",����Q@;�75��LP",��6/*CMa1t���� �>.",��6?��*CMB6/*C>.",��6/8A>)�.",��7w��6/�5�275�B3�+.�a ��2:<;o�� �>)��^/75Q$�2;
RS>@�,�4 �>@��^/75Q��2;��275���b35��6/�5�275�4 �*=Q@ <;��6�*CM@�."$>�R i 35��6/�5�27A <: j 35��6/�5�27A �*���6��@:���;*$+�-�^?��%z:<1_M)� i j 35��6/�5�27�3���14��Q@;�75��L,�=6/*CM@�6�*CM@��"$>�R i 35��6��5�27�1B���2 ���:<;��P��>�6�6?�2:�� j 35��6/�5�27�1=���2 ���:<;tZ�>���6?��QAQ,��L$MA>P�275��� i j 35��6/�5�27�3���1���2 ���:<;����5��%���-k ¤"$��6?'[� j 35��6/�5�275���$�2 ���:<;=�5Z�>���60��Q,��10���!75�5>�6/*CM��275��� i j 35��6/�5�27�3���1.��Q);<75��L,�
R i 35��6��5�27�1
34
McCulloch & Pitts ���¥�,�,����¦ �,�5�,�5�
RG��60>!75��1Tv�L�Q@75+) �>@%p35��6/�5�275�P �*~�@QA*,;��¡'[��^?+ (MP) McCulloch & Pitts *§ -?��+,�R�¢ H R c NGL,+D60���5>@£����� �>b����L,��75%/�4 �*=�,��6?>a75�� ��?�@ < �%0��*$+D�)+,��%/Lb:^/1?+D35��6/�5�275�
Σ
Wi1
Wi2
Wi3
µ
© ª<« ¬ McCulloch & Pitts �®°¯�±5²<±5³´±´µ[¶B· ¸<¹ ª º2»$¼ ²±´³[½¾ ",�5>!1?�t",��1?^?+A�b��75*$+D-/8A>a1��$�����a � ���%/�_35��6/�5�275�&6?��+$>)%
( H R c ) Vthe cell is activethe cell is passivei =
¿ÀGÁ 10
¾ �fgh:��a7w���@7�-/-0�21�35��6/�5�27_ �*9�AM)��+@75��-/�( H R H ) V k W V ki ij j i
j
+ = ⋅ −
ÂÃÄ
ÅÆ�Ç∑1
È É È�Éθ µ
¾ �fgV:��A6/-0'f��+D6/*C>b�A'[6�-/+D�,����^_Ma7w�2;��!75��� θ �q���2 ���:<;��tvkL=�275��� µ i6/*C>)%
( H R U ) θ xif xotherwise
È�É=
≥¿À Á 1 00
R�¢ H R c NGL,+D6?���5>b�A>)6/£�35��6/�5�27� ��$��LA��75%/�4 �*s � �%?��*$+="$��%�LD�fgh:�3��,��7 i 35��6��5�275�r �*=Q@ �M)�r �%\�L
( H R W ) V k W V ki ijj
j( ) ( )= ⋅∑�a ���-0',35��6/�5�27� ��)L,��75%��D6/*C>@%��a ��2:<;��� �>@��^/75Q��2;� �*B��6��a�a75���D35��6/�5�275�b �*p�,��-?��LA���D�a7w��%/�,�
RS",�5��L,+=vkLA+( H R j ) W V kij j i
j
⋅ − ⟩∑ ( ) µÊ Ë
0
35
¾ 6/*C>)%�! k�2:<;��r k>@��^/75Q$�2;�����6/^_�������a �vkL µ i
i ��Q@;<75��L Q,L��2;��b35��6��5�27� j ��Q);<75��L �@6�;�35��6/�5�27�3���1���Q@;<75��LA�b6/*CM)�b�A+,^_�2: Wij
�-0��+T�275���.",�a �*��A��1?^?��:��=�,*$6��s �-0��+,* Wij��%�6k:\ �*.�)+,�5>)�,+B�)6���8!1#�fgh:�8)��%?���a o��8)�� �^?�
MP
`�6/-/�D8)6�%/�a14�275�5>b�AZa �-?��+BR F�ceYKJ �$����LA��L@1?�&�$����60>!75��1?�&�$�2 ��2:�;��r �%� *s:���^?��1�6�*,;�>@+��q�� L,6k10��7w��>RS�2 �>D",��1_��*$��84:���^?��1/ ��)8A�275�,�?�)6/���)+,�
�����,*$6� �*T",�5>a1?�p",��6/Q,+A6k;��=�$>=35�21?*$8a1&>@��10+.�275�5>,�_6/�$����1oQ,*,;k��+s �-0��+B�275���D�AZ@ �-0��+¾ ������1?^?��:
>)^_��+@1.�5'f�2 .g c gz �*~�a:��21_M��)+A6.>! ��x�a ��2:<;=�� k>@��^/75Q$�2;Ì �*~",��;�^?6=6/^?��+����,��+�>�35��6��5�27 •
RG35��6/�5�275�4 �*=�,>a �+,�&�,�� �+A*$8)�b�$�2 ��:<;< ��a ���M@*=6/*C>35����%�+��$����7w��6�%k75��L�3�75�5>������5'f�2 �����1?�P�����,*$6���R�",��756/%�7w��L,+¡"@75�5>��A���,��+$>P�,*$6�1.",��7w��6��5�275� •
�A>@6�£$��Q@;<75��L�3���1?�P8A����6���+@ =",��6�Q,��+AL@756/Q ��6/��7�6?��6/8)��*��fgV:b:�^/10�,+¡",�5>@�,�P3���1p6/*CM)�)*RG��7w��6�%k75��L$>A�q�5>)6?MA>D�275���D",��6/QA��+,L)756/Q ��6/��75�b60��6/8)��*=`/�� ��)�BR�¢ c R U vk��:�L
RG�)�AZ,�27�75�5>D",��7w��6/�5�275�4 �* Propagation Delay � •
RG-���-k1o>a ��mv���^?6� �-?��+D�a75���&�,��1?^?��:�b�,*$6�� •
�����,*$6D �*��,������%/�5>P�a7�1?�a D`�6/-=�,^?��6k;t�A������� MP *� �-0��+,�T��%D3�����/ r1q��*$8�6�1?-b �*¡��+$��%���L@ R�v�LDQ@75+) �>@%\35��6��5�27m �*=�,QA*,;���+,�&�A'[^?�)�b�fgh:��$>AZ��q������1?^?��:
Hopfield ���¥�,�,����¦ �,�5�,�5Í
ÎÐÏwÑ)Ò
�A6?Z[:<1#",��1?^/:��D�A%/6k:+9���2 ���:<;\��>s����8)�@75+=6/*C>D�$�21?6��$��-0�21/:��$��'[^_��+=�$�27w��6�8,>)�=",��75*,1�fgh:=���2:�^_�21?+A*~�$����^_M!7w�2;����%B�)8a75�)���A-/+��2:p��Z]�)*$�5'���6?��8A>)+ÓRG�,��QAL,��QAQ,L��AM@��75%�+� �*�����Q,��*
R F�ceYKJ 6�8,>D��>b�AZ)-/-?�21�35��6��5�27m *B>a ��q�A��1?��Q�M! ���Mb���2 ���:<;� �*=6/^_���="@75���D",��10^/:��b�A%/6�:�+�a:��5'´;�6�^_�������,*T>! �1z1?^/:��5>)� 10 104 5− %�:�^_��+,+)1&"$���)��-/+.",���A+B6?'w�214",-0>t �*¤��8,��+)1
RG�$����10^?��:��b�����,*$6��r �*p¢O�$��6����,��£ Redundancy �b�)7w��%/�=>@���t��Z(R�",-0>@�&-?��M!;���@1��,��-���+ �*= ���-?')6k;�LA+,+=�$�21?%�6?��+A�D�$��%/6�:�+,%o",��1?^/:������,*$6\8)�@7� �6/*,;k>@+D�,��QAL,��QAQ,LD�AM@��75%�+@1�*C��+,��*$�RG�,*$6��r �*p",����1?��Q�M! ���M)�b����'[�a75�,�)�b��7�;���>b��>b3���1?�a ��?�,��-�-/�b������6?��*$�5Mb�� <:�1o",��Q$��*,;�",��Q@75+) �>
36
35��6��5�27� �*y�$�2 ���:<;p3���1T�a <1_M)�@�¡�$>~��>@6B6�*C>�",��7w��*C>@6���",�5>AM@�5Zf��;���3���1.�)����-� ���;<;���� �*¤",����6/*,;�>)�=",��1?^?+,�D��75*p6?��+$>)%rRS",����Q@7w'[+p",��6/+���8! Ising �*= �-?��+)1o",��75��;�L,�&�$��'[�a75�,�a �-/-0�21
R Si = −1 1,k p ¾ ",�&3���;�L,�",��75��;�LA��60>@*$+P�A>@^_���A%
i3���;LP*$8@*��� �MA�2 ����)-/*$�PR J ij %&�@75+$��LA+P",��75��;�LA�.3���1D�)��^_M@6�Q@75�5>@�R j Sij j
j
⋅∑ ¾ �275���b�,%�6k:�+)1R h i
o ¾ �fgh:�3�+$��LA+ i �b3���;�LA�r <:� <:<;k��+D6/*C>b��7w��^?��8)�b�)-/*$�¾ �fgh:�3��,��7 Si Vi
3���1�6�*CM@�&6?��+$>)%( H R � ) V Si i= +1
21
Ô Õ
"$>BÖ c �)���)�m3���;�LA�r10^?+D6?��+$>)%( H R { ) J S hij j i
j
⋅ + >∑ 0 0
RG�a ��2:�;��r �>)��^/75Q$�2;�6/^?��+D6�*C>b35��6/�5�27� � ���M)*9�AZ!10^?+
¾ 60�21/:( H RX× ) J S hij j i
j
⋅ + <∑ 0 0
R ���:<;�>! �35��6/�5�27!'[^?��+D�AZ!10^?+ <:o",�):<;�*$�a ��m�A*$6k1#",����6?��QAQ,��L$MA>@�A�a",����60��Q,��1?���a75�5>)�=",�5>@�,�D��>s������*$�a \3��,��7!6?��+$>)%�A+,^_�2:< � ���M@* µ vkLA��RG����-/*� *��,��-�-/�B�)��^_MA>@6/Q)75�5>.",��6/^_���)6/*C>9�� �� �*C�a��1_����8=3���;�L,%z�a10��1?L,�
Re� i �&3���;�L,�4 <:� <:��2;�6/*C> hi0 ��7w��^?��8)�b�)-/*$�
�-0��+�RS",��1?^/:\�����,*$6�8A���,��7� ��,��QAL,��Q,QALD�AM@��75%/+)1�*C��+,��*$�&�$>b�)6/*,;��5>)*B>)���D��Z)�! �1_M@� �*Ø�,��10��Q$>@��^_��L�>Ù�!;�� �*C�b",��10^/:������,*$6k1§35��6/%/�5Zb �*n�$����:�1/ ��$��LA8)���,�Ú6/*,;�>)+Ù�AZsQA*C�2;�+
R FiH<{KJ_EGFiH��KJ �,*$6/�)+D���27w��6/%?Z
Û�ÜDÝ_Þ�ß)ÜÙÞ�à\áhÒ�âhá�ãaÞ
������+$��M@+ �$�21?��^?�ä�$��-?��Ma7n�@ �:<1å�A%/6�:�+,%æ",��1?^/:�� �A%/6�:�+ ��> 6���-?'[� -� ���;<;�����A%/6k:+@1§",����6/*,;�>)�Ú",��1?^?+,�Ó %çR (state space) 1??+,�y��75�,*$+Ó108@6�+@1 (local stable point)35��6�%/�5Zf�è�$>è�)*,:�+@ é�,�����)+ê�$��L)75%/�,�)�Ø�,-?��Ma7�R (attractors) �$��L@75%��,���Ø�$��-?��Ma7� é",��L@75%/�A+
37
�fgh:Ú35��6�%/�5Z[1Ø",���5��'[*ë",��Q,6k;é35M@�) ��B",�� �*$�a ��a ���%��.��Z��,%�6k:�+�RG�A%/6k:+,�ì �*í��10��Q$>)��^_��L$>@�¾ �fgh:��A6/-?'w��+ N -/+A��+@1��)-/��-�1��,%/6�:�+BRG���21?��^?�_�,-?��Ma7� ��,%/6�:�+,�&�$��L)75%/�,�
V V V V VN= 1 2 3, , .....î ï
�a �8)�,�)�P�5>a75�P6?�21/:&��%D8)��7�7�RG�,%�6k:�+)1=6/+,*,75*�35��6/%��5Z��275��� V V V V VN0
10
20
30 0= , , .....
ð ñ ��%D8)��7�7R V0 �L)75%/�,�D�A%/6�:�+,�&3�Q$MDM@��;L,+ ∆V 6?�21�:��5Z�> E V V V= +0 ∆
∆V 6?�21�:z�A%/6�:�+,�9 *��)�5'f�2 ��2;���QA��RG35��6�%?Zf�B��1_'� z�5Ma �89:�-���+���'[^?��+ V �a �8)�,�)�B�,-0��M!735������%�+Ó��1?��Q�>@��^_��L�>ç35��6/%��5Zz�!75���y��Zz�,%�6k:�+ÚR���>,�2 +@1P35��6�%/�5Zf�~�$>~6?Zf8)*, 9 �%0���y",��75Q$M~M@��;�LA+
R V0 �$>DMA����-k1�6?Zf8)*, �3��,��7 V �5M! 8r:�-���+D6?��Q$MA����+,*L)75%/�,�a ��,%/6�:�+,�=v�>@*$�p",�)�� �>s6�*C>9",��60��Q$M)6/Q$>D6k;L,+=�@ <:<1��AZ�>!;�b6?��*$��+)1��,�� �M)�5Zf��;��A%/6�:�+
RContent-addressable memory �$>@60M!7E ",��7w��6��5�275�P�,LA��75%k =",��7w��6��5�275��>@^_��+�3���191q��*$+P��6/*CM� < k��%D�$����1?^?��:&�����,*$6k &-� ���;<;����. �*� -?��+,�6�1?-B �*��2;k��L@1T�,*$6����$>~>@��10�~6/*C>��AM@��+@75��-.�A8@�);��,+~�A*$6k p�@*$-�8��)LA��75%.",�5M!;�LA+�6/*C>)%
R�¢O�AZf%/*D1?^?+=",���5MA�q�)-/��+)1��$>AZò£<10��^?�q10^?+@ ¾ �fgh:�3��,��7 i j >@�D3���1��)L@;<75��LA�b`�6k:
( H R Y ) W V Vij is
js
s
N
= − −=∑ 2 1 2 1
1
Ê Ë�Ê Ë
��Q@;<75��LD�)6�;��b35��6��5�275�r �*=�$�2 ���:�;��b�A+,6 Vi��Q);<75��LDQ,L��2;��b35��6��5�275�r �*=�$�2 ���:�;��b�A+,6 Vj
��Q);<75��L,�&6/*CM@�&�,+A^_�2: Wij
��>���6?��Q,QA��L$MA>@£B��;���>ó���$��>��� <:<1ê"@75��� j � i 35��6/�5�27°6/*C>)%ô�a ��2:ä��Q@;�75��L,�í-0��+,��^?�í`�6k:RS¢O��6?��QA��1?���a75�5>
¾ �fgV:��a7w���)7 Hopfield *s �-0��+,�4 �*9�AM)��+@75��-/�( H R ced ) V sgn W Vi ij j
ji= ⋅ −
ÂÃÄ
ÅÆ Ç∑ θ
¾ 6/*C>)%i 35��6/�5�27� j 35��6/�5�275�&3���1�6/*CM@�&�,+A^_�2: Wij
��Q@;<75��LD�)6�;��b35��6��5�275�r �*=�$�2 ���:�;��b�A+,6 Vj
��Q);<75��LDQ,L��2;��b35��6��5�275�r �*=�$�2 ���:�;��b�A+,6 Vi
i 35��6/�5�27_ �*=�$�2 ���:<;� \v�L,�&�,+,6 θ i
38
�)75Q$M@�a ��a;��5>)*$���fgh:��):<1_Ma7 Ising *� �-?��+A�A� Hopfield �*Ù�A*$6/�Ó �*Ù�AM)��+@75��-/�R �,%/6�:�+,�4 �*=�@�5'56k7w>@�
¢ H R c<c £ E W V Viji j
i j= ∑∑≠
12
RG35��6/%��5Zf�@+b:-/��+b �*=�a;��� �*D:�^/1?�A�b��%\6/+�>! �3��,��7��5Z�> ∆E = 0 6/*C>)%",���5�275��*$� E ��7w��6/%�75��L$>õ35��%�-/��:Ø",��:^/1?+u�27w> E �,*$6� n��7w��6/%�75��Lö35��%�-/��:Ù:�^/1� Ø"$��M@+)1min +���>@/ o�,*$6� z6�*,;�>@+.*,:�6��@*T35������%/+T",����Q,L)75��+,6�Q,-/�¤",���5�275��*$�B��7�;� �:4",��;���-�:o",�5>@60M,>@�
RG�� �>AM,�2 �A+,6.�$>~",��75%/-�:�+�R�",�5��L,+~35��6��5�27� �*y�,�5>@60M,>��)6/��8a1T�fgh:p�,*,:<7��,*$6��� *ç�AM)��+@75��-/�"$����L)1 E ",��75%/*$�T",��7w��6/�5�275�B �*P���2 ���:<;��.�A+,6k1t�$�2 ��A%D¢ H R H �)8)L$�27$;?gh:£ i �.35��6/�5�275�B �*P���2 ���:<;��
RG�a ��� 8b6?Z���8&35��%/-/��:�,�?�,�5>)6?MA>b�@60��^/1ovkL��27a35��6��5�27)",��6�8,�21�35��%�-/��:�� ��6���75��6?��6/8)��*ö �*u�A��7w��%k75��L�>@�A���,�5>)6?M,>)���$�2 ���:<;�õ��>õ�);�M@*$+���Z��,��7w��6�%k75��L$>í���2 ���:<;
RG��Q@;<75��LD3���10�b8A����6/+)1o",��6/QA��+,L@756�Q�A��:<1� �:\6k1q'[�,�a ��,6/*,;k>@+D �-?��+A�b �*.��Z,�a7w��%/� E ��60>!75�� �>! � �-0��+=�275���b-� ���;<;����4 �*s -?��+,�35��6�%/�5Z# *�*C��+A��+ÙR�����-/8)���$����7�1?��6�;�L,+~ �*Úv/��6�^P3���1P-���6k;��a .�)*CM@�A+Ó6�*C>ç��6?>a75�� ��~35��6/%��5Zò� <1qM!7 E S S S= +0 6 0 451 2. . ¾ �a75���~�)L,��75%/�~�A��7�1?�Ó6�*C>@% E ",����*CMa1� kM@�,��7���6?>a75�� B��10��Q$>@��^_��L�>60��6/��1/14:��21qM! o3��,��7A-k ��;<;����D �*B �-?��+)14�)Q@ �8)�)�Bv�L@14*C��+A��*.�fgh:4R O O O= +0 6 0 41 2. . ¾ >@^_��+)1
R O2 ���> O1 �6/�����q1q��6?M O "$>@�
39
������������ ���������������������������� �� !�
Hodgkin & Huxley �� "��������� �� !�� !#
$�%'&)(*%+$�,-$�&/.10+243�,657&628.19:&-0';�<=0'$�</5?>@%+<A%CB!0',-0+5D3�E),=%GF'0'3�E=HJI!>10',�5D9/5?>@%+<�,5LKM%CNL2PORQTSMU�V�WRQTS�X�V�WRQTSMY�V�WRQTSLSZV[F!%CB\%+0'>],^5?9_0!E`B'KD,a24(1KD,a240'3)2cb�0'0!%+3�0+B�$�.1>]3�<d%C5L2eE)$�,[9d$�%+0+B\%+0'3�0+B!,b�0!&6B!$)2f%+9�KLBgb�0'0!%+3�0+B!,�ORF'.1$[B!,�%�F!I'5D9:&),h$�%C5iKD$j5D9_$�%+0'<)B!0'>],j$�%CB\%+k1$=,j$�&j$�%+.@&)$=<l$�%'&�%'%+9�<�,$�%+<�.�0+B!9lF'0+2m,=In.1>]<h$�.@%+(12mo�NME�%+,Pp[$�<=,Pb�0'0!%+3=0+B!,a$=.1>]0'3)2qOsrVoltage Clamp t p[$�<aK]%C240!Eu5D9
O $�%CB\%+9vp)$=<
wxzyz{*{6|Gw}x�~����Osr�SMO�Sg.4%+0!& t 0+5D<=9�pqI�%C5LB\&G$=.@�nKL2�.4&)$)5}F'$=0+B[b�%+.4E),G$�&
RK = 1/gK RNa = 1/gNa RL = 1/gL
Ek ENa EL
���z�
���z���
Cm
� ��� �+�6���v�)�!�M�q�?���n�i��� [�q¡��'�D¢n�[�P£ �L¤¥� ¦C§��v�?�¨�n�F'<��\2©b�0'0!%C5D$ g ª gk Na− %C5D0!&�%«b�0+K]%C2*EPW R E E E Cl l k Na m, , , , b�0+240'k1.1<=,G0'k�&[0',G,[p6B!,),
Osb�%+.@E[,Gp[$�<[2*%¬ 0��K®b�0+B!$=0+B�5D>4%+<�,�5D9Pb�0+B\%+0',cb�0'<�.4�z,
r�SDO�¯°S tI g V EI g V EI g V E
Na Na m Na
k k m k
L L m L
= −= −= −
( )( )( )
40
,65LKD$=,8OR0+ND0'(1ND37F!%+0+5h$�%+>DK]%+0'<±.]9:&²$�%C5iKD$)2³b�0'.M2*%CKjb�0+B\%+9�,±b�0'<=.@��,´0'klp[0+B!<²5?>@%+<�,.]9:&)k�Wµb�0+B\%+0',¶5?98.M21K?<±.19�NM&)$�<´,[p=%+$)Nf,65LKD$�,·.19:&)k_OR.4%'In37%'&¸p�%+$[Na24(@<)2d$�%+0',65f,65]%+k10.DND3�<[2�,)0!%C5?$8F!I¨5D9:&�%¹F'.1$[B'5ºb�%+.@E),´5D9»$�%+k10+5M%+<�,�O�b�0+B\%+0',²5D9».M2]KD<�,¼b�3�p6B�,[.4%'In38,65LKD$�,
OR$�%+p=%+$[ND,G$�%C5LKD$�,
~µw¾½¿��À«x�Á[y�Â�{`ÃÄ�}Å[y�Å`½¿��w®Æ�y�Ã-Çex\ÈÉw
,=E`NM%+3´%C5]&¸b�0'0!%+3=0+B'2_OVoltage Clamp Ê 2 b�0'0!%+3�0+B�5?9 ,[.1>10'3·%CKD(@0+2f0+5D3�E[,=%ÉF'0!E�HsIn>@%+,O�b�0+B\%+9Pb�0+5D>4I¨2�,6B!.M24<�<�,mp[$�<)2�,=In.]>1<q$=(@0+N]E
,[>10'>]<PË1.DKD<P.4&�%+$�<^SMO�Y}HR3�<P.4%+0!&`2qORoik@%60+5D3�E[,jW F'0!E�HsIn>@%+,v0��K�,65L5Dk4%+9:%«5M%'Ea0��K�,[p[$�%CN�%'��,[;�0'9O (Voltage clamp) p[$=<�,ÌKM%C2@0!EÌ5D9
+
-
V
I
Í+ÎsÏÑÐ�Ò+Ó'Ô
Õ'ÎJÖ×J×sÔ'Ø
ÐCÙ ÚnÔ!Í�Õ+ÎsÖ
Ò'Ó+ÔÍ+Ø ÎJÐ¥Ô+Ô!ÍÐÑÛ ÜÝ Ù ÜÑÏ'Þ¥Í
Þ+ÓßÍZÕ'Ù ÎsÏ�à�ÎJ×\Õ'ÎJÖ
� áLâ � Voltage clamp �D�_��ã@�!ã1���qä4�?å4�P£ �L¤Ñá ¦Ñ§��v�?�¨�n�
41
5]%+<_p[$=<_$�>@%+.];�E65]&)kq9�<=9�<_>1p=&),�Osb�0!E)>væD3�kq0';�%+pd0+B!9¸%+3[B!k@%+,dF!%+3�E�&[,P5D9�%+k1.4%'&`5b�.4�\5m5]%'E)9¶b�.@��,j2e%+9�<)2fb�.@��,�Osb�.@�ç$�>@%+.];�E65]&)kq9�<=9�<_0+B!9�,l;�%+p[,=%�WR$�0+B\%+(@0'p[,l,[>4%+.1;�E65]&),
ORF!%+3�E�&[,Ì5D9fb�%+.@E[,GË1.]>¹.M2e%CKD9a0+B\%+0',
Voltage clamp ) p)$=<�,aKM%C2@0!Ea.D2*In<[5�,)3=0+B!kM2�,[.4E 2@,vp[$�<P$�&Pb�0+KL2e%'EaF!%+9:&).],u2]5D9�2ORF!%+3�E�&[,Gp[$�<qo�NME�%+0*,)0+5M&Gp)$=<�,m$�In.]>1<q$�<=(*%CK}$)KD0+2eE`5},65]%'E)9v,[.@E`24,Gp[$�<q$[KD0+2*EaO�r amplifierp[$�<=<¸,6B\%+98W Vm
,6B!.M2@<=<�,�p[$�<�9´F!%'%+0'k]<¸9�.MND,[,�.M2eIn<)5v,[3�0+B!kD2db�0'p[$�<¸9�.DND,�9�0¿,=�¿2]5D9�2b�.4�z,_5D97%+k1.DKL5G5M%'E)9¸.]9:&�b�.4�p)$[ND$�0è9�.MND,[,_.M2eIn<·&)(*%+<)2^b�0'p[$�<�9�.MND,l2eE`KfO Vc
,[.4E`2@,0+B!9-F'0+27b�0'p[$�<é9�.MND,éb�0'0!E�>@%CK_5?k´b�.@�vb�0'.@��0�9�.MND,[,».M2eIn</O F!%+3�E�&[,8$)B!.D2@<�<»Ë1.]>�0+B\%+0',
OR.M2eIn<)5}$�%+3�0+B!k1,,[>10'>]<�,¸Ë].MKD<[2�OR0+2*%+0'p´b�.4��kj,[(*%+p[,¸,[<�3[5LN Ê %+3�E�&[,)<7&[(*%+0',´b�.@��,¸$�&¶F'<=3)5v5L2*%'E[<
OR0+2*%+0'pc%CB!0',Gp[$�<=,ÌK]%C240!EG.M2eIn<�<v&[(*%+0'9Pb�.4�z,O −15mV Ê 5^,).4E`24,²$=3�0+B!kdp)$=<�<7$�&²,6B!9�B W −70mV %CB!0',7&)$=,¸5]&)0'(1B!;�%CNP0'kh,[<�I�%+>D5Pp[0+B'B
Ê 5}KD0!In0*,6B!.M2@<=<�,mp)$=<�<q.]9:&G>MK®&)$[50+2e%+0'pqb�.4��b�0'.@��0*.M2eIn<�,m0'k¹,`N?(1B Vc Ê <qF';�E Vm Ê 9aF!%'%+0'k]<O −15mV
Os&[$�,qË4%+$)5¹F'.]$)B!,vb�.4�)5D>@I6Ë1k]<q,=&)(*%+$�k4%eF'.]$)B'5}$�%+k]0+5]%+<)2�,65D>4In,65},�&[0+2@,m,)0'(*��0'.D5]%CND>1,5]&[0'(1B!;�%CN�.D21KL5�KLB�,[0',h&[$�,Pp[$�<=<�%[,`5M%CKLN�5M&)0'(1B!;�%CN�p[$)ND$=<h,)0',fp)$=<�,aKM%C2@0!EP$�k].MKD<h.1>DKD,62Ë]kvb�.@�E6ND3�<_p[$=<�,jKM%C2@0!El.M2eIn< Voltage clamp 5D9_,6240+2@3)2ub)5]%'&³W F'.1$)B!,P5D9l5ME)9�<�,h0!%'%+0'9
O KM%C2*EG.@&[9�B[b�%+.4E),mp)$=<�9b�%+.4EjË1.]>G0+B\%+0',db�.@�n5�5M%'E)9_p[$�<=,^KM%C2@0!Ej.D2*In<dË1.]>Gb�.4��, Voltage Clamp Ê ,PË].MKD<[25D9dp[$�<h$�(@0+N]EP.4%C21K�F!I¨5D9:&),=%[F'.1$)B!,v5D9l$�%+k10+5]%+<=,P0'0!%CB!0'9l,[<�I\%+>M5�.@&�%+$=<dSMOsS�H�3=<j.4%+0!&`2qO�&)$�,
OR,[p=%CB!<�,�5]&)0'(1B!;�%CND<hê 45mV
� �°� �+ 6�����Gå]�'ë4�n�v�n�=¢¨��ëìß¡��D��[�)�@ì!�L�:¢n�G�=�'í4�n���'�v�L�'�6�h£ �L¤Ñî ¦Ñ§����L�!�n�
42
ï y���x�y���wx\|[y�{�xz½
ð\ñ {`ÃGÈÉ{mwx\|[yz{*x�½ç�
¬ 0�òK.4&)$[5F'$=0+B6F!%+3�E�&[,qb�%+.@E62�F!I¨5D9:&[,G$�%C5iKD$G5D9v$�%+k10+5M%+<�,G$�&
r�SMOó¯LY t g g nk k= 4
F!I¨5D9:&`5®b�%+.4E),Ì[email protected]'<=,G$�%+k10+5]%+<=, Ê gk
F!I¨5D9:&`5}$�%+k10+5]%+<=,G.@%CK?0'9a$�&ÌKL2*%'EG.]9:&qb�0'>1<�0'<q.]3�pm,`B!$=9�< Ê n0!��& n=o .19:&[kd&)<�I\%+>M5dOsôMO�O�mF'0+2ºb�0+KLB�%+0'k].MK]%�F'<��\2e%�p)$=<)2h$�%C5D$�kf$�%CB!$�9�,65q5]%+k]0 n ,6B!$�9�<�,$�&��ÉW g gk k= 5L2*E6B n=1 .4%C21K�OR$�%+.4%'In3a$�%C5LKD$=,G5Dk19PF!%'%+0'k�F!I¨5D9:&qË10+5M%+<P&65�b�%+.@E[,G.]<�%C5Dk W gk=0
OR$�0+5?<�0'3�E[<v&)0',mF!I¨5D9:&`5}$�%+k10+5]%+<=,�%@$�%+p=%+$[NF!I'5D9:&),m$�%C5LKD$G5?k19v$�.1<�%'&b)B!9�0�$�%C5LKD$=,8Ë@%+$)2´O 0'.1k@%+3 Ê 0+B\%C215?p²2@k].1,7$�%C5LKL2�$�%C5]%'E`5M%+<�<8$�%+0!%CB'2�$�%+0+B\%+0',±$�%C5LKD$�,0+5D<=9�p[,a,[>19�,=%6p[$�<[2�$�%C5D$=k�b�<�%'E[0'<P$�&fb�0+B!9�<P.19:&vr�b�0'.MK?9�,v%C5]& t 0+5D<�9�pqF+KD;=<a0+5iKL2�b�0+B'24<
Os&)$=,qb�%+.@EÌ5iK¹b�0+5iN]%CB6.]9:&b�0'.DKD9PX0�òK$=9�KLB«F!I¨5D9:&),m$)5LK?$q$�p[0'$)ND9f&�%+, n4 .];�<=.MNL5}0'.M5M%'E`5]%+<q0+5]E[0!��0+ND,G9:%+.]0+ND,HR3�<v.@%+0!&G,=&). t F!I¨5D9:&),m.M2]KD<q$�&ÌK]%CB!<[5}0'>1kM2�.4%'In3q,[0',[0�>]p�&G.DKD9�9fE)0+ND3=<POsr�SMO�U HR3�<q.@%+0!&G,=&[. t
OsróSMOsõö=÷¹ø·ù[ú�û
ü ª ÷
ýþ�ÿ��
ø����¶ý ª ù��K
���û�ø ���:ú��( ��� ª ��� )
ì+�ߧ����[ë¹�� 6�¥����ì'¡��D��Ì����å1�f�L�'�6�h£ �L¤C� ¦C§��v�?�¨�n�OR,65LKD$[2�F!I¨5D9:&),m.M2]KD<q$�&ÌK]%CB!<[5}0'>1kM2�b�%+3=pG>]p�&G.DKD9fE)0+ND3=<q.@%+<�&[k
43
ü ª ÷
ýþ�ÿ��
ø����¶ý ª ù�� ý ª�� ÷´ù[ú�û
ì+�ߧ����[ë¹���L�'¡�§qì'¡��D��Ì����å1�f�L�'�6�h£ �L¤Câ ¦C§��v�?�¨�n�
KD0+N?9�<P,=��,)>]9lW�24(1KD,G2@0'3[5¹ËDB\%'&)<)2�0+5D<=9�pq,)>]9P.1(*%CB`;�9�ND$=<a,65]%CKLN?,G5]&)0'(1B!;�%CN¹.19:&[kO�rJ$�%C5]%'E65]%+<vb�,qb�0'.DKD9�, t b�0'.MKD9�,�5D9Pb)24(@<G5LK
¬ F!%+9:&[.¹.1>13=<q$�0'<=0'k¹,)0'(*E=&).¹0��K}.4&)$)5}F'$=0+B6.4%'In3)5}p=%+$)Nç2@(@<=<q.MKD9�,m.M2]KD< r�SMOó¯�X t n n����� ������! "�# n
n $% &(')α
β
← → −1*
OR.@%'In3[5}p�%+$[Nç2@(@<�<q.D21KD<=,G$�%+.10',[<q0+K]%C2*Eqb�, α βn nª−
¬ 0��K}F'$�0+B«p=%+$)Nç2@(@<[2�b�0!&)(@<[B!9Pb�0'.DKD9�,[<vE`5Dp62�0!%CB!0'9�,Ì24(*E r�SMOó¯LU t dn
dtn n
n n+�, �-�.�/103254,%� n+�, �6�.�/(�67983,%� n
= − − ⋅ = −∞α βτ
( )1:<;= >= ?b)K©r�SMO�¯iU t ,=&�%'%+9�<PF!%+.]$)N�0!��&PrJ,=In.]>1<P$=(@0+N]E t $=0'<�%'&)$=0+N�,[.@%+(12�,6B!.M24<�<=,ap[$�<=<P$�&fb�0+B!9�<hb�&
¬ 0��K}F'$�0+B«,6ND9a0!&6B!$ r�SMOó¯°õ t n t n en
n n
n
n n
tn n( ) ( ) ( )=+
−+
− ⋅ − + ⋅αα β
αα β
α β0
r�SMOó¯.@ t n t n n n et
n( ) ( )= − − ⋅∞ ∞
−
0τ
¬ .19:&[k r�SMOó¯�A t τ
α βnn n
=+1
44
r�SMOó¯.B t n n
n nn n∞ =
+= ⋅α
α βα τ
OR>]0'<�$�<=,Ì24(@<)2�b�0'k1.DK®b�, β αn nª− .19:&[k
���' 6�¥���*�G���ß��å1���DCn�' [�v£ �L¤FE ¦C§��v�L�!�n�b�0+2e%+0'plb�0'p[$=<)2PO $�%+.@%'I!3lF!I¨5D9:&[,h$�%C5LKD$h2*%+.mF'kM5]%�>4&)<lF';�E n∞
>@&[<lb�0+5?0+5D9_b�0'p[$�<[2O $�%+p=%+$)N®%+0',[0�F!I'5D9:&),m$�%C5LKD$G2e%+.�%C5]&qb�0!&`B!$[2�F'kM5]%6¯ Ê 5ç[email protected])$�< n∞
O p)$=<q$�<=.5DkM2�b�0'0+B!0+N]%'&qb�0+KM%C2*Eqb�0'k1.DK}9�0 β αn nG
− Ê 5}0'k�%+p[0+B!,G0+5?3�E),=%@F'0!E�HsIn>@%+,0+NDk�F!I¨5D9:&[,8$�%+k]0+5]%+<±5D9»$�0+B\%+0'3�0+B!,7,[<�%'E`K?,±.@%'&[$)5 n Ê <8.]$�%+0�,)<=0!&)$=< n4 ,[<�%'E6KD,±$=.@%+(
OsrDSMO�YèHR3�<q.@%+0!&G,=&[. t p[$�<GK]%C240!EÌ5D9Pb�0!&6B!$)2�,[>1>]<)B!95L2eE)$=,)9 n Ê 5h0!%+;=0+2@,´$)24(@,²0�òK_OsSMO�¯iYq,[p)3�%CB'2_,6B\%+$)B�F!I'5D9:&),¸5D98$�%+k10+5]%+<=,¸$�<�%'E`K
¬ F!I¨5D9:&[,G$�%+k10+5]%+<[5¹&62@,m0!%+;�0+2@,m$�&Ì5L2eE`B`rDSMO�¯.@�,[p[3�%CB t p[$�<�,�K]%C2@0!EG0!%+3=0+B'2 r�SMO�SLô t g g n g n g n n n ek k k k
t H= = ⋅ = − −
IJK LMFNOP1Q R ST∞ ∞
−4 1
44 1
40
4U V W Xτ
45
¬ 0'k�SMOó¯.B�,)p[3�%CB!<�%12@0'(@0',Ì24(@<)2�SMOó¯LY:,)p[3�%CB�N@òK¹b�%+9�.D5}F'$�0+B«.@%+<�&[kr�SDOsSM¯ t n
g
gk
k
n
n n∞ = YZ[ \]_^ =
+∞
14 α
α β>10'<�$=<�,�2@(@<)2�F!I'5D9:&),m$�%+k10+5M%+< Ê gk∞
F!I¨5?9:&`5}$�0+5D<=0'3�E[<q$�%+k10+5M%+< Ê gk
>]0'<�$=<�,Ì24(@<)2�$�%+p�%+$[ND,G$�%C5LKD$=,Ì5D9v0'3�p[0',qE65Dp[, Ê n∞
Ê 2cSMO�SM¯,[p[3�%CB�$�&a2@0'(1Bµb�&lO n∞5D9º%+k1.MK�$�&a249�p`5�Ë1k�.]p�&=%)$�%+.10'9�0[>@%+>]<)5�F'$�0+B gk Ê % gk∞
$�&
¬ 0'k5L2*E6B[SDOsSLô
r�SDOsSLS tg t g n g n g n
g g g e
k k k k
k k k
tn
( ) ( ) ( )
{( ) [( ) ( ) ] }
/ /= = = =
= − − ⋅
∞ ∞
∞ ∞
−
4 1 4 4 4 1 4 4
14
14
14 4
0
τ
OsróSMORX�H�3=<j.4%+0!&P,�&[. t F!I¨5D9:&[,a$�%+k10+5]%+<[5�$�0+B\%+0'3�0+B=,[<�%'E6K�,65L2*E[$�, Voltage clamp 5D9h0!%+3�0+B[5DkD2OsSDOsSLSl,[p[3�%CB'2´,=In(*%+,[9�0+NDk�$�%+k]0+5]%+<�,8$=p[3�%CB'5_$=0'.1;=<�.MNº,[<�&)$=,8,[$�9�KLB�$=0+B\%+0'3�0+B!,8,[<�%'E6KL5OR,6B!.D2@<�<=,±p[$�<[2·b�0'0!%CB!0'9�,±0'.]p�&=%0+B'NL5h>]0'<�$�<=,¸2@(@<)2 ,6ND980'k1.DKdb)B!0', gk∞
Ê % gk 0
.@%+<�&[k
n∞ Ê % τnË1<�3v5LK�OR$�0+B\%+0'3=0+B!,a,[<�%'E`KL5�.]$�%+0+2mb�0!&[$�<�, τn
F'<���,GK]%C2eEc5L2eE)$=<a$�%+>10'>1<=,qË@%+.DKD0'9�<O 240'(@0',Ì2@(@<[2�p[$�<�,m$�<=.¹.@%C21K βn Ê % αn
0'k1.DK®b�0+2@9:%+p[<r Vss
p)$=<�< t 240'(@0',Ì24(@<)2 αn$�&Ì2@9�p6B SMO�¯.B ,[p[3�%CB*5LK},)0'(15M%CND0+B\%+<q0��K
r�SDOsSMY t ατn ss
n
VnW X
= ∞
¬ r Vssp[$=<�< t 240'(@0',Ì2@(@<[2 βn
$�&Ì2@9�p6B SMO�¯�A ,[p[3�%CB*5LK},)0'(15M%CND0+B\%+<q0��K]%r�SDOsS�X t β
τα
τ τ τn ssn
nn n n
Vn
nW X W X
= − = − = −∞∞
1 1 11
$=0'.1;�<=.MN�,[<�&)$�,h0��KcOsb�0+B\%+9 Vssp)$=<�<d0'k1.MK©.@%C21K βn Ê % αn
0'k1.MK©%C5L2eE)$�,^b�0'0!%+3�0+B!,P$=.1>10'3=<OLSDORX`B Ê %*SMORXaA:,[p[3�%CB'2�In(*%+<=9a0+NDk¹p[$=<)2�$�%C5?$�k βn Ê % αn Ê 5®b�0'0!%+;�0+2¹5L2eE`B
46
ð Çew¾�z{mwx\|[yz{*x�½ç�
,[0'(12@0';�E�& ¬ b�0'k]0+5D,[$´0+B!9±0'>10ç5LKf$)KL2eE`B}F!%+3�E=&),·b�%+.@E`2^F'.1$)B!,�$�%C5iKD$�5D9´$�%+k]0+5]%+<�,¬ 0��K},`B\%+$[B6F'.]$)B!,m$�%+k10+5M%+<q.@%'&)$[5},�&[(*%+,[9a,=&�%'%+9�<=,aOR,[0'(1240';�E�&6B!0!&�%
r�SDOsSMU t g g m hNa Na= ⋅ ⋅3
F'.1$[B'5}$�0+5D<=0'3�E)<q$�%+k10+5]%+< Ê gNa
ôDO�O�¯:F'0+2�b�0+B!$�9�<=9Pb�0'.];�<�.DN Ê m,h%+0',[0 m 0'.DKD9l$=9:%C5D9�9lË].@%+(G9�0�&)$=,hb�%+.@EPË1.]>�F'.1$)B!,f0+B\%+0)5D9d.M21K?<�,P$�&^.]9�N]&`5�0'>1kD2OR$=0'.10+ND<=0!&±,[.@%+(12�,[9�KLB�b�0'p=%+$)ND,7b�0'.DKD9�,´.MND3=<±$)K?0+2*E O p�%+$[NP2@(@<[2_,[0',[0 h .DKD9:%¹b�0'p=%+$)N$�&jb�0+Ki2*%'EP.19:&hb�0'<�.4%'In,A5Dk�5D9l$�%+0!%+.M2@$=3�,[,P$[5LNDk1<^&)0',v,)p=%+$[N�,)0',[$P,65LK?$�,[9h$�%+.M24$�3=,),
O m h3 .1<�%C5?k�W�,65LK?$�,G$=p[0'$)N¯¹2@(@< t p�%+$[N h .MKD9·%C5D0!&�%:b�0'.@%'I!3 m 0'.DKD9_5]E)9�<=,l0!%'%+0'9_5]&)0'(1B!;�%CNL5�2*%+.@Edb�%+.@E[,jp[$=<l.19:&[kOm 0'.DKD9h$�<=0'3�pq2*E`K�F'.]$)B'5�,[.@%'I!3P,65LKD$�,vp=%+$)N h .DKD9�9dæ1&u5iK�,)p=%CB!<=,ap[$�<[2qOsr�SDObA}H�3=<P.@%+0!&62Ë]0+5D,[$l%+,=�¿W�b�0'p)$[NLB m 0'.MK?9l$�9:%C5?9:%)$�%C5iKD$�,PË].@%'&65�0+5D<=9�p[,P,[>19�,P,6B!$�9�<h,[0'(*��0'.M5]%CN?>Ì.]p=&`5%+,=�ÌWR.4In3�,65^b�0+5D0'p[$�< h 0'.DKD9´$�k19�<[B},)0'(*��0'.D5]%CND,[>1,[9�klWsróSMObAmHR3�<¸.4%+0!&`2dSÌ24(@< t ,[0'(12@0';�E�&),m 0'.MKD9�,[p=%CB!<�,±5M&)0'(1B!;�%CNi5�b�0'.@��%+p8.]9:&)k´Osr�SMObA^H�3=<�.4%+0!&`27Yq2@(@< t ,[0'(12@0';�E�&6B!0!&),±Ë]0+5D,[$HR3�<v.@%+0!&62�Xç2@(@< t p=%+$)ND,G24(@<)5 h .MK?9a.4��%+pm,[0'(12@0';�E=&`B!0!&[,)<q,[<[5Dp[,),m.1p=&`5M%e$�%+.10',[<)2�b�0'.4In3)B
OsróSMObA
47
c%d_ebfgih%jbelk�hnmcod_ebfgih%jbelk�h%m M3
Hp.hnj!d H
M3
Na
1 2
Na
3
M3
H
M3
H
4
Na
c1qrg.j�c(esh g5t3u�gc(qrg.jp�ebqwv_xc(ely�v�z.{
cod_ebfgihn|.h%jsdp�elqwvz�g�{
{i}�elt.}M hnqru�}
H qui}frm5z{�t3u�g{�|�eby3m {�t3u�g{�m3elgid
start
ì!�?�µ¢n�G���°å1�v�D�_�����n��¢��!ã1�qä1�i���P£ �L¤9~ ¦C§��v�?�¨�n�Y�F'0'(@< M
Y Ê %�>1p=&�.DKD9´F'0'0'(@< M WR>]p�&�.MKD9´F'0'(@< H Os&)$=,¸b�%+.4E¸%CB!0',_.@%CN]&[,_.@%'��&),O�b�0'.MKD9
¬ b�%+9�.D5F'$=0+B[róSMO�¯�Xç,[p[3�%CB t F!I¨5D9:&[,Ì5D9fb�0'.MKD9�5},[<�%+>D2
r�SDOsSLõ t m m+�, �-�.�+�, 03/(254 +�, �-�.�+�, �6/(798m
m�α
β
← → −1 ?
r�SDOsS�@ t h h+�, �-�.�+�, 03/1254 +�, �-�.�+�, �6/(798h
h� α
β
← → −1*
¬ .19:&[kp=%+$[N¿24(@<)5}.@%'I!3G2@(@<�< m Ê % h 5D9v.M2]KD<�,m0+K]%C2*E Ê α αm h,.@%'In3G24(@<)5}p=%+$)Nç2@(@<=< m Ê % h 5D9v.M2]KD<�,m0+K]%C2*E Ê β βm h,
48
OR$�%+0+B\%+0'3�0+B!,G$�%'&)(*%+$[5$�%'&�%'%+9�<=,G$�<�&)$=,q&)0',m0+5D3�E[,=%eF'0!E=HJIn>],Ì5D9v$�%'&=%'%+9�<�,m$�.1;=<¬ 0�òK,6B\%+$[B m Ê % h 0'.DKD9q5D9a,=E[0'<)B!0'>1,
r�SDOsS�A t dmdt
m mm m= − − ⋅α β( )1
r�SDOsS�B t dhdt
h hh h= − − ⋅α β( )1
¬ 0�òKF'$=0+B\%�SDO�¯LU:,=&�%'%+9�<qF!%+.1$[NL5},[,��=SMOsS�B Ê %*SMOsS`AÉ$�%'&�%'%+9�<qF!%+.]$)Nr�SDO�Y°ô t m m m m e
tm= − − ⋅∞ ∞
−( )0
τ
r�SDO�YL¯ t h h h h et
h= − − ⋅∞ ∞
−( )0
τ
¬ .19:&[kr�SDO�Y°S t m m
m m∞ =
+α
α β
r�SDO�YLY t h h
h h∞ =
+α
α β
r�SDO�Y�X t τα βm
m m
=+1
r�SDO�YLU t τα βh
h h
=+1
p[$�<=,h$�(1N]E[,j.]p=&`5�24.GF'<��çb�0'p=%+$)N?,lb�0'.DKD9�,a5D9ºb�0!E`5Dp[,hb�, h mG∞ ∞−
.19:&[k,[$�9�KiBd%CB!<�0'<=9 p[$�<)2Äb�0'p=%+$)ND, b�0'.DKD9�, E`5?pA$�& b�0'.@&)$=< h m�
0 0− O�b�0!%+3�<AË1.DKL5
¬ b�%+9�.D5}F'$�0+B`SMOsSMU Ê %*SMO�YL¯çWiSMO�YZôg$�%'&�%'%+9�<=,G0��K�OR,[(1N]E[,),
r�SDO�Y°õ t g t g m m m e h h h eNa Na
t tm n( ) = − − ⋅
IJK LM N⋅ − − ⋅IJK LM N
∞ ∞
−
∞ ∞
−
0
3
0
W X W Xτ τ
49
$=9:%C5D9�.4%+<�&[kPO m∞$�<�%CKL5�p[0+B\� m0
F'kM5mb�0'.4%'In3 m 0'.MKD9�<h5]%+>@I�E65Dpdb�0+5D0+5D9�b�0'p[$�<[20'k].MKm$�0'.4%+;���0'.M5]%CN?>qp)$=<_$�(@0+N]Eh$)KD(124$�<d.19:&)kvORF'.1$[B!,P5D9_,[0'(12@0';�E=&),h$�&db�0'.@&[$�< m 0'.MKD9,65]%'E[9P.19:&qF'.1$[B!,q$�%+k10+5M%+<)2�>1.@%+0',q,=��&`N?,q$�&q.@&)$=<a>]p=& h .MKD9f.@%+<�&[kÌW h0
$�<�%CKL5�b�0'p[0+B\� h∞
OR,[0'(12@0';�E�&`B!0!&[,GË]0+5D,[$)5¬ r�SMOóY°õ t ,=&�%'%+9�<q$�&G;=9�NL5}F'$�0+B
r�SDO�Y.@ t g t g m h e eN N
g
t t
a a
Na
m h( ) ( )'
= ⋅ ⋅ −∞
− −3
031: ;= >= τ τ
p[$�<vK]%C2@0!Ea0!%+3=0+B'2m%C5L2eE)$�,[9h$�%+<�%'E`KL5�$=0'.1;=<�.MN�,[<�&)$�,v0��K¹5L2eE`5�F'$=0+B τh Ê % τm ′gNa0'k].MK¹$�&
ORp[$=<q$�<�.5?kM55]E[9�<�,P0!%'%+0'9h5]&)0'(1B!;�%CN�,6B!0',[9l$=0+2@.]0'<�,Pp[$�<=,P$�<=.M5�p[$�<aK]%C240!Ea0!%+3�0+B�b�0+KD(124<P.19:&[kb�0!&[(@<)B m 0'.DKD9±2*%+.ºb�0+2*%+0'p7b�0'p[$�<[2@98F!%'%+0'k1<»W m∞ = 1 0'kj5L2eE`B W 55mV &�%+,[9 Na Ê ,¸5D9
OsrJp=%+$)N}.DKD9 t ,).]0'$�<=,G,[>1<[KL2
6���=ë���'�°��í¹ë@�6�@�n�që��1��ë¹���1�¨�=�'�=�q��ã1�[å@ë m �\�?å*��q�=�L a£ �L¤!� ¦Ñ§��v�?�!�!�
50
O Na Ê 5}$�0+24.10'<�,m$�%+k]0+5]%+<�,m$�&G>@%+>]<)5ç5Dk4%CB6,=�)0!%+3=0+B'2r�SDO�Y�A t g
gm h
gNaNa
hm
Na= ′⋅
= ′∞
==∞
30
11
0
In(*%+<=9-0+NDk Na Ê , 5?9/$=0+B\%+0'3�0+B!,»$�%+k]0+5]%+<�,»$�<�%'E KL5�0'.];�<�.DN_Ë4%+.MKD0'9�<85L2*E[$�< ′gNa.@%+<�&[k
OLrDSMOóY.@ t ,[p)3�%CB'2¬ N@�K},[>10'>]<�,G$�%'&)(*%+$=< m∞
$�&Ì249�p`5}F'$=0+B6;[KDkr�SDO�Y.B t m
gg
Na
Na∞ = ′
3
b�0'p=%+$)N h 0'.MKD9»2*%+.�>@&)<éb�0'0+5D0+5D9Äb�0'p)$=<)249-&µ°�aW h0 1= >4&)<»b�0'0+5D0+5D9éb�0'p[$�<[2�.@%+<�&[k.]$�%+0'9´F!%'%+0'k1<�,65]%CKq,[0'(12@;�E�&6B!0!&),_$�<=.ap[$=<�,_$�<=.a$�&�b�0+5iKD<�9�5?k1khOR,6B!;�E_,)0'(1240';�E=&`B!0!&),=%
Osrs3)N]&65¹æ]&�%+9 ht b�0'.@%'I!3 h 0'.MKD9
6���=ë���'�°��í¹ë@�6�@�n�që��1��ë¹���1�¨�=�'�=�q��ã1�[å@ë h ���?åe��q���L v£ �L¤!��� ¦Ñ§��v�?�!�!�5D9éF'<���,80+K]%C2*E8Ë@%+.DKD0'9�2·,)9�KiB!9�0+NDk·E�%+0'>M2 βm Ê % αm
F'<���,80+K]%C2eE±$�&²5L2eE`5dF'$�0+B¬ rDSMO�SMY Ê SMO�S�Xç$�%'&)p[3�%CB6,=&). t F!I¨5D9:&),
51
r�SDORXDô t ατm
m
m= ∞
r�SDORX1¯ t βτm
m
m= − ∞1
1W X
O βh v( ) Ê % α h v( ) .]9:E),G$�&GF'k@%@>1.DNLB'2�p[$�<q$�<=.5DkM5 βh Ê % αh$�&Ì249�p65F'$=0+B6Ë]k
0!%+.4I�b�0+5L5]%+k�.19:&qp[$�<qK]%C2@0!Eq0'0!%+3�0+B«5D9P$)N?3�%CB[,[.1>]0'3a0�òK�&�%+(@<[5¹F'$=0+B62@0'(@0',�2@(@<)2 h $�&$�&�%�2@0'(@0',j24(@<)2 h $�&h>@%+>]<)5�F'$=0+Bè%C5]&lb�0'0!%+3=0+B'2PORQÑS°X�V[F'p=%C2@,h0!%+.@I:0+B'NL5�p[$�<d$�In.]>1<j5?9_,`B!$�%+<
¬ F'<��z,m0+K]%C2*EG$�&q&�%+(@<[5F'$=0+B60!%+3=0+B�5?kM2�WRF!&[k1<PO τh
r�SDORXDS t ατh
h
h= ∞
r�SDORX1Y t β τhm
h= − ∞1W X
βh Ê % αh5D98$�%C5D$�,´$�&¶5L2eE`5hF'$=0+B�$�%CB\%+98p[$�<7$�%+<�.d.@%C2]KhF'<���,¸0+K]%C2eE�5DkdæM3�%'&[<
ORp[$�<[2
¬ QÑSDU�V¥&`24,G$�%'&�%'%+9�<�,m;�3q0��K}0+5D3�E),=%@F'0!E�HsIn>@%+,�5D9u5?>@%+<�,m$�&qb�k]3)5}F'$�0+Br�SDORXLX tI C
dvdt
I Cdvdt
g m h v E g n V E g V Em ions m Na N k k L La= + = + − + − + −3 4( )
W X W X
r�SDORX1U t �n n V n Vn V n
Vn nm
= ⋅ − − =⋅ −∞φ α β
φτ
1� � ��� ��� ������
r�SDORXDõ t �m m V m Vm V m
Vm mm
= ⋅ − − =⋅ −∞φ α β
φτ
1� � ��� ��� ������
r�SDORX`@ t �h h V h Vh V h
Vh hh
= ⋅ − − =⋅ −∞φ α β
φτ
1� � ��� ��� ������
0��K}F'$�0+B\%@0!%+3�0+B!,G$=.@%+;=.MND<=;)2�0!%C5D$q.]9:&ÌK]%C2*Eq%CB!0', φ
φ = −3 6 3 10( . )/T
OR3�%+0'3)5D(�$�%C5iKD<)2�,[.@%+;=.MND<=;�,m,`B!0', Ê T
¬ b)B!0',GF'<��n2�&65]%ep[$=<)2�b�0'0!%C5D$q.]9:&qb�0'.1;�<=.MND,=%
52
r�SDORXaA t α n VV
V
W�X W XW X� �=− −− − −
0 01 50
0 1 50 1
.
exp .
r�SDORX`B t βn VV��� � �
= − −IJK LM N
0 12560
80. exp
r�SDO�U°ô t αm VVV
��� � �� �W X= − −− − −
0 1 350 1 35 1.
exp .
r�SDO�UL¯ t βm VV��� � �
= − −IJK LM N
460
18exp
r�SDO�U°S t α h vV��� � �
= − −IJK LM N
0 0760
20. exp
r�SDO�ULY t βh VV
��� � �W X=− − +
10 1 30 1exp .
r�SDO�U�X t n vV
V Vn
n n∞ =
+��� ������ ���α
α β
r�SDO�ULU t τα βn
n n
vV V
��� ��� ���=+1
r�SDO�U°õ t m vV
V Vm
m m∞ =
+��� ������ ���α
α β
r�SDO�U.@ t τα βm
m m
vV V
��� ��� ���=+1
r�SDO�U�A t h vV
V Vh
h h∞ =
+��� ������ ���α
α β
r�SDO�U.B t τα βh
h h
vV V
��� ��� ���=+1
53
������������� ��������������� �����
� Simulink �������� "!#�$�%!'&)("*,+.-/�10#2)3"45�.-60#-"7�-/8%*%!#*$0#9)7%:��1!<;=0�:?>@4�A6!#�B9)8�9CA"!#*?�
�1D Icon E?0=FG�,!'2H3I8J+G LK 9)A"!#*1�M9)8N�?4�("�G�O�%+�("*G�QP �SRTR >U7?*V3"!#0�+�- H � H 9)8 Icon ���M0=36!#F?+�*
4�(�!#*18W0#2) LE?0=*?36DYXZ��*[�G-6 \36!#*W]SP �U^_^ >`71*[��FG7,!'��a?9)A"!#*1���1F�7%!'� � 9)A6!#*?�V9)8WE?0=b?3)2)�NE?0#9I4��1*
� P �SR Pc>U7?*H3"!#0�+�-
Simulink d e fTgihkj?l5mChkn�oLp qTr'sTs tvu%j5mCh�w�n
x qTr'y_y tvu%jHl?z�u?h�{�l1n?m)|�f_gih=j1lH}?h�mT~�p qTrvs)q t'u%j5mTh�w�n
E?36D��M9C8WE?0 - icon - ��*�AIF1+�9) H0=36!#F1+G* � �13\F1+M�1:G0=*G��0=AH����8%0�E?0=*?36D���*VAIF?+�9C i9
� E?3"D��<9)8H9)A6!#*?�H+�("*G�
54
9)8��?:�0=*G��0=AI�G*5�G&_2)8J!#*��1D%E?36D.9)85�1:G0=*G��0=AI� � ;�4=9)8J+G��E?3"D.9C8�9)A6!#*?��4�(�!#*5+.-6��3"!#0�+�-
� n �����?8%*?�
x qTr s)q t'u%jHl?z�u?h�{.l1n1mi|"�=�f)��n�l5�%mi�%p qTrvsT� t'u%j5mCh�w�n
� +.-6��3"!#0�+�-��%4�(�!#* n �����18%*�9)8��1:�0=*G��0=A\�
x qTr's_� tvu%jHl?z�u?h�{�l1n?m)| n mC����p qTrvsTy t'u%j5mTh�w�n
55
9)8O�1:G0=*G��0=AI��*���&T2)8J!#*L�1D�E?3"D%9)8M�1:G0=*���0=A\� � ;=3\�G����E?3"D?9)8�9)A"!#*1��4�(�!#*�+.-"�53"!#0�+�-
� h ��! m E?0#���?8%*?�
x qTr�q�� t'u%jHl1zGu?h�{�l?n1mi|"��mToJ{�l5�%m)��p q_r'sT� t'u%j5mTh�w�n
� +.-6��36!#0�+.-��%4�(�!#* m �����18%*�9)8��1:�0=*G��0=A\�
x qTr�qT� t'u%jHl1zGu?h�{�l?n1mi| m mC����p q_r'sT� t'u%j5mTh�w�n
56
� +.-6��3"!#0�+�-��%4�(�!#* h �����18%*�9)8��1:�0=*G��0=A\�
x qTr#q�� tvu%jHl?z�u?h�{�l1n?m)| h mC����p qTrvs)� t'u%j5mTh�w�n
� +�-"��3"!#0�+.- 4�(�!#*�&I!'-�:����14�0#9ID��HE?36D�9¡9CA"!#*?�
x qTrvyTy t'u%jHl1zGu?h�{�l?n1mi|�l?�¢w�fT�£l5�%mi�%p qTr'sC� tvu%j5mCh�w�n
�,!'9T-�:��?* 152
µAcm
9)8¤E?3"D��,4�3\AI*¥�%!'&)("*,+.- HH 9C8¦9CA"!#*?�§�%+¨E?0=3i&)3)&)*©3\8J+�
� 9)A6!#*?�<9C8���0=(\9I!#*10=7?���,!=+G(�!#�H�,!=4�(�!#*5E?0�+.-6�HE?0=3"!#0�+�- � +G�1��F��?*�-��%!#0=(\9)0=("7,!=+
57
152
µAcm
f)�NmCh�mIw�� mChkei� x l,w�ª\f�hkj%w�u%|�l%{£mie\j1j?lHz�o%j�p q_r'sT� t'u%j5mTh�w�n
n w�mT���NmChkei��l%w�ª\f�hkj%w�uHo?hknGª)h#oLp qTr�q�« t'u%j5mTh�w�n
58
m w£mC���NmChkei��l,w�ª\f�hkj%w�uHo?hkn�ªih#oLp q_r#q�s t'u%j5mTh�w�n
h w�mT���NmChkei��l%w�ª\f�hkj%w�uHo?hknGª)h#oLp qTr�qTq t'u%j5mTh�w�n
59
Fitzhugh -BVP����������� ��� ��� �
���������������������� !�"�����#�����%$'&����(�)�*&�+��#�%�-,���./+0�21%���3���(���!�3��45��$�,#6!�Fitzhugh
��,������,74819�%��:#�7������ "$;��1%:���$9$!$<�=���(���!�3�>+?�@$��7,�+��!�3��4"& �BA/CDA ���E,���1<$'$;13�����7�)�-F%,#�7�)�HG7��,#:I1J�
� $��#,�+����?$;��CK�)A $!�J�L$������ 1<$!��>+I,����#�;�;�(:#+��M1%$N��4O$;�������!4;F<�!�P$����������7�Q$;�BA/���#$!�%�R�<�>+S$��
Fitzhugh��4N�S�(�����
�������<�',���F)�#�T� 1%�#�U����$���+>�U�@V�WYX[Z;& ��,#� �BA �Q$���CK�)A $��Q$�1N,�1<$<�I,(����1N,7481N6!4�\7���]��6!��$!�N�������$�1N4;�!��4
Van der Pol��4R��,�1JA ��� 1J�������������R� ��,(��_%�N�������<�U_)\�$� !�J��� ,74�\71<�'���#�T6!4�\7���
� ��:#+a1<$H��4T���(���%�L ��� <���Fitzhugh�
Van der Pol��4b��A $����R�>+c � ;�B���
BVP-Bonhoeffer - Van der Pol��4d�S�(�����
e �gfh+L�)A/��$<Ai��,(1)A �����H��A $��!�H$;1!����4;�j X��lk>mon p@p pV KV V+ + =φ 0
e �!1)�(�H�%,���:(�<�T�%���%$T�!1�����4;�!�H��,(1)A ���a1J���?�<,�_<�%�*�B��� 1��j X��lk�W[n p@p p
V V V V+ − + =2 1 0q r
φ
e ���=�;��4;,�A/�(�!1�����4;���2�>+L�%��:(,(CK6%A � 1s+�:#�#Aj X��lk>Xon pV
VV W+ − + =
3
30
�@�!.�1)\��H,���4;���<�*Cg��:(���L&�$!��A �H$;��1!����4;�tCg��.J�0��_JAi& 1<�7�
j X��lk�u[n ppV V
VW
W V
= − −
=
3
3φ�@�!1J���!�
BVP$;1�����4;�T$�1T�;��4;,��L&�$���Ai�%�(�B�7+��H$!�(��_JAi$;.v.g�!�w��A/��:(��F?,(��,7��+x$%\� ;���?�gfh+
j X��lkoy>n ppV V
VW I
W V a bW
= − − +
= + −
3
3φ z {
e ,#481%�����+��B��_T�<A ���T|a b, ,φ�)A ,��������?F%$����t|
V
60
�%�<��F%�%�H�)A $�4;�t|W$��#,�+��<�L��A/��:(��F?,(�B+���+a��,(.!|
I
e ��� 1)�(�I����:��B��� 1<�I����� _<$��)�a�����#��,7:&��=���!�
1 23 1 0 1
12
− ⟨ ⟨ ⟨ ⟨ ⟨ }~������b a b bφ
,��B�#+'�@���#�>�'$!F!1]$���,��7��CDA �� R�%�(��_)Aa�<+t�J����:(��$��#,�+��]�>�w_<$!$I � mH,(�B�#+S4�F%��6%�(�)�T���7���U$;1�.v�
�@$���,���_<���2�7�B\L,#��& 1!�w$�1�����4;�T$;1s�0��_JAa=b=z=0,7F�1)�2�(��:(���9��:(�<�?�%,(.gF!�`�)A ,��(���!�9��4M,(��,7��+��'$<A/���7$�$;1'$�,74�\71<�
W��4Q�!_%���<A ���#�
,���� 1J�2$�CK:����Simulink
|D� j X��lkoya�� !�3�%F% ;�BA%�!1%,#n��S�(�����H�S4E�%��:#�7������ !�T48�������'�@�� <�7�B\S��� 1T� ��,#� Ce 1J�(�
Simulink � � Fitzhugh - BVP �S�N�>�����!�E� �>���o� ���;�t�S� � �
61
e 1)�(�H,���� 1J�=$�CK:����T�!CK,#�7�T,(��,#��+0���S�(�����?$<�w��CK$
� �����>� �7�D�(� BVP ���N�0�����!�H�;���� g�'� �>���Y¡ ���;�t�S�D�K�
�@�%��:#����6;_�1%�d�)A $�4;�b$�1d,(1%$��W
�)A $�4;�!�!�¢�)A ,������!�dF%$��£$;1�,�1<$!�V�JA $!4;�
��4",(��,7��+-,��B�#+*F<$!�<�?$;�����BA $'$<�>�w_c,74�\71<�'�!.P,��(�T¤V�)A $�4;�%�-$!�� �F%�%��.K\>�s.�.v���
W�)A $�4;���
����,(.<$�CK,#�7�$!�(��_JA��t$!,(.v��FM$��#,�+����!�¥�)�7�B+>\T�71%��:#A 6;�B\tF%$<\S$��¦1)�U��� �� ��¦§� ��R&�6�_]� ��,#� CK�M,#481%�|K�T& ��,7$���\a$;1t��� 1���,*X��lX�¨©,(��� 1J�?�@�)�7�B+>\��71%��:#A 6;�B\L,#:��BAJ_<��\S ��H�7����CJ,���,#��+��?,#481%�-� �7_<4;�!�H� ������4
�l§� ��%�T�!�B��CK�?��C ,#�#�T,���,#��+>���>�w_<$!�T,#481limit cycle
62
V � W ���g�)ª��t�S�D���K�����0�����!�H�; v«#�E� �0���o¬ �B�;�t�>� �K�
�HH
��4����������"�����?$;�B+>\7��$!�E���H$�19,(1%$<�-��Cg�� ��5�BA � 1���� _J�SF��BA ���BVP
��49�S�(������71%��:#A 6;�B\=��4'$��7,(�����t��.v1)\��%n(&�,#$<A �T��4'�%,#���%���t�%��:#�(��6�_�1JA � 1%�3$�13,#���� ��)���7Cg�� !�'1J���S�(�����
�@�)�7�B+>\S�s�71%��:#A 6;�B\���4U�����(F%,=���� <�7�B\a�����>�w_<$!�T&����7� j �)�7�B+>\��$�����:#����:( ;��1%�,��( ��)�M��$����#� 1 !�� <�O�������%���V�W>WhZ*¤@V�WYm[Z
Fitzhugh�gfh+QF%$;�BA
BVP��4�S�(�����
� �S�(�����2��4t�<�%��F%�%�H�)A $�4;�H��4t�!_<���%A ���#�H�gf®+x�)A ,��(���%�
63
Rinzel����������� �� ��� ��
������������������� � ���"!��#�%$�!"�&�'�����"!��)(�*���+����������"��*-,.���/(�0�*21435�#�"* HH �����76#�"*��148#*��9!"$�����*:��+�0�*;���=<�!�(���* Rinzel �7�:��64�"*��;� 1#8#*9!"$�����*9,?>�@?��3)�A@�*��B�7�:���C!�*�$�!�65� �EDGFIHKJMLEDN,�OPJQL DN,5RSJ)T5��@����UT5!��)8.>��������V>�!��5(�*��A+����A+�0�*W���X<�!�(���*W<�3#�M>�!�!"$�������Y!�0�ZC!��/�'��ZC�/[ n\ n <�!"1?���C!�]^��*�!�!����%�)�_@`+��a�)��!�85$�(��_+�TA���C*�1?!�3W<�!�Z%14� Fitzhugh�%$�����*/�5!�64bc$d�fef<��5��$��-�C�`@��-���9�C!�851#!�(����%$�!����-�%$�����*/[ h\ 1-h <�!"15�&eg���"Z��"��+��h<�bc���������_�i@��
�H ��65ZeNj��fkAF \ H h= −1 , �E6)Z��%�U0�+)�X<�!"1 h �"*�3d�56#bl�"*��?�'$�����*?���"� H� ��!������)�XT)�56d�A1m�_@�� L �)��!��W���'�n1m�"�#���&���o�`1M�C��* n-H ���?��]p�%+��X�4�"��!�*�1q>�!"$�$l�_14��*W�5���C3
r !tsS@u�������������"��*W���W>��"���A$eNj��fk`k \ H sn h= = −1 �'14�WvA�4bc1w�#���"��* n [t� H <�!"1w�5�����
n x y H z�{}|w~)������� �`�G��� �_���?~`y�{}��E�CZ��_$�*�1 n [t� H !�3)�A@UT5*�0X�`@U���_�)�%1d1#��Z%�U<���!"$'�����o@A�_+�!������
64
eNj��fk`� \ s Hn= 0
0
<���!"$ dmax
���V�"35��@w�f����������!"14�5!�*��W�_��64��b'���X�#�C��*W�5��� � d
max@�(���*W�5!�64bc$ L �)��!��W�������`@�
{n Hmax max
, }[ �
{HALfH
}���"64���'$���<�!"1w!�65!"�)�������W�CZC�5*���!tsS@
eNj��fkAR \ d n HHn s
smax max max
/max
max
= + ==
= +2 22
11
11e j
r ��!����C�5�C*?��ZC�5*��X���X�#����*W�5��� W��*��#�_$�*W�%$�����*W�)!�6#b}���)]}$�!��
eNj��fk`O \ W dd
d
s
i e Wsd
s= =
+�������E� � �� =
+max, . .
11 12
12 2
12
� �
r !ts@�<���!"$%��!������)��*W�C64���'$��W�CZC�5*�L L ���C�o�`@wL (n H, ) ���V>�!"$�!"+)���W>�!�35��@U�5Z%15$eNj��f�`H \ d n H n n s
n s
s
n sH
s= + = + =
+
+= +
+2 2 2 2 2
2
2 2
1
1 1
� �
�����)*W�5��� W �7*��#�_$�*��X�%$�����*��X��Z�0��_$%���o�`1M�%$MjA��k`O �CZC0��_$¡T#�"�C�ujA�f�`H¢�CZC0��_$¡���o1#!�85$¡�5���C3 � H [c� n <�!"1w�)�����X���
eNj��f�A£ \ W s1 s
Hs
1 sn
2
2 2=+
++
� h [t� n >�!"$�����*��&>�@U���"!��#�%$�!"�U���������"��*W!����¤�7��(�0W�����"�C*?eNjA�fkik \ �������"��*��MeNjA�f��£ \ �������"��* � W ¥ H !�3d�����C�A�U<���!"$ h [c�U!��"(�!"14�C*�� h W= −1 [ � n
WS
= r !�3��`1m�'$ W ���V>�!�Z'$l�"*C1 �W ��*��4�_$�*����'$�����*�������>�!"+A�"�58w!tsS@ h [l� n >�!"$�����*��X���WvA!"��Z%$l� HH ���������"��*��U�%$�+ $
65
r !tsS@U��$���!"$ H [l� n >�!"$�����*����������C!�*�$�!�65�eNj��f�`j \ dndt
n V nVn
= −∞ ¦¨§¦©§τ
eNj��f�Aª \ dhdt
h V hVh
= −∞ ¦©§¦©§τ
�%$l�"*�1w6)Z��X�50�Z%$l�)vA!�0��_$�LE6)Z��X0��_$�!�*�1d��!"+73A$ r jA�f��ª«�CZC0��_$����`@U���"!�85�)�_+7!"$�*W�A+�0�*X@�8515$ r !�3��`1m�'$l�eNj��f��, \ d h
dth V h
Vh
1 1 1− = − − −∞¦ § ¦©§¬¦ §¦©§τ
r !�3w>��"�����U<���!"$¡�'�A�����?���4�"851M�eNj��f�AF \ dHdt
H V HVh
= −∞ ¦©§¦©§τ
r !�3��`1m�'$'jA�f�A£��CZC0��_$¡���W>�!��#]��b%�5���C3eNj��f�`k \ dW
dtS
SdHdt
SS
dndt
=+
⋅ ++
⋅2
2 21 1
]���QjA���`j¢��ZC0��_$�1d�`1M�C���C��!��"(�!"151 dndt���Wv�!"��Z%$l� j��f�A£ �CZ�0��_$�1d�`1M�C���C�?!��"(�!"151 dH
dt���WvA!"��Z%$ r !�3��`1m�'$
eNj��f�`� \ dWdt
s s H V H
V
s s n V n
V
s s H V H s s n V n
V
h n
=+ −
++ −
=
=+ − + + −
−∞
−∞
−∞
−∞
2 2 1 2 1
2 2 1 2 1
1 1
1 1
� � ¦¨§¦©§� � ¦¨§¦©§� � ¦©§ � � ¦©§¦©§
τ τ
τ
r 65��+`$�1w�%$l�"*������o1#��Z%$
66
eNj��f�AR \S S H V H S S n V n
S H V W S n V nS
S H V W S n VWS
S
S H V Sn V S W W
S
S n V SH VS
S W
S
W V W
2 2 1 2 1
2
2
2
2
2 2
2
2
2
2
1 1
1
1
1
1
1
1
+ − + + − =
=− + −
+=
=− + −
®¯° ±²"³+
=
=+ − +
+=
=++
−++
=
= −
−∞
−∞
∞ ∞
∞ ∞
∞
∞ ∞
∞
� � ¦¨§ � � ¦©§¦©§ ¦©§� �¦©§ ¦©§� �¦©§ ¦©§ � �� �¦©§ ¦¨§� � � �� �¦©§
r !�3��`1M�%$l� 1− ∞h va f 1#!�85$'e jA�f�AR �CZC0��_$�1 \ H v∞ a f>�����*C1
eNj��f�`O \ W VS n V S h V
S∞∞ ∞=
+ −+¦©§ ¦©§ ¦¨§´ µ� �1
1 2
<���!"$%<�3��U6#��*?>��C!"$�!"1q>�!"1m�"�#���5��64��b'�)650?�"������* τh va f [ � τn va f <�*�]t�X!"@)�_1m�X���������U<���!"$��?!"+�3 � τ va f �A�����?<�*�]'@A�_1M�X!tsS@u>�+�!"��ZC�%�eNj��NRIH \ τ V V¦©§ ¦ §= − + +5 10 55 12 2exp / ]
r 14!�85$l� jA�fj��N£ vA!"@�0C1q��b}8M�"�C��!"+�3 HH���?���"!��#���C*������������"��*C�U��@�3d�4��]¶Z%$<�3A� h [t� n ���?<�*�]t�o@A�_1M��*X�76#��b%�56)0�1w<�(�� m ���?<�*�]t��@)�_1M��� m v∞ a f 1#!�85$ m t v,a f >�����*C1 � £<���!"$�L m �'$�����*�1q���"0�@`�u>�����*C1w<�3A�)�#6#�C*W�5�C*X1#!�8#!#1#8#*C�n@�!�b}* m �%$�����*���!�3d�5*��'�U<���!"$� m v∞ a f 14!�8#!��o1#8#*C1 m
�%$�����*C1q�'+���0��%�n
WS
= >�����*C1 � j h W= −1 1#!�85$ H h W= − =1 >�����*C1 � ª
� @`14�#�W>����C*�1d1#8#*W!"$�����*W!"$���>�@
HH�������"��*X�7����(���+)�"*W�C0��)!�bQ� 1M�%$%���_1#8#�C�o@)�"8#!"1w�)Z��'�eNj��NR`£ \ dW
dtW V W
V=
−∞φτ¦©§¦©§
eNj��NRIj \C
dVdt
I g m V W V V g ws V V g V Vm Na Na K K L L= − − − − − − −∞
3 41¦©§¶¦ §¶· ¸ ¹ º&· ¸ · ¸
67
� Simulink [}�X�C$�3#�"��1 Rinzel ���W��6#�"*��X���"*�!�*?b}8M�"*?�'14�X�#�"!��%1
Simulink x | Rinzel »��-»`¼Ay½���?�^y½��{}��� �`�G�`¾ �_���?~`y�{}�
¿ �`�"À`� �Á���W��ÂC��y�Ã%����~�Ä V ��ÃcÅ������W»��-»`¼Ay½���?�^y���{}��� �`�G�`À �_���?~7yc{}�
68
¿ �i�"À�Æ �_���W��ÂC��y�Ã%����~AÄW ��Ã�Å������W»7�B»`¼�y����?�^y���{���� �`�"��Ç �Á���?~7y�{���ijA�NªIHnÈE0�*W�#�"!��%1w��b}8M�"* 15
2µA
cm
�%1M��bc1q>��4]���b}�56)*X�����4�_@��5!"@U�4�_15@U�C!�85�)�"*�!�0��X�����C8M�"�
V Â%Å����W��Ã�Å����?~7y½|AÉw��{}Ê5»�y½��{}���XÅ�y��CÊAyGÅa� �`�_ËiÌ �Á���?~7yc{}�
69
W ����»iÂ����W��ÃcÅ ���?~7y½|AÉw��{�Ê5»�y½��{}���XÅ�y��CÊAyGÅa� �`�_Ë`Æ �Á���?~7y�{��
limit cycle Íty½���Xz"y�~`Å�Îq»��-��Ï}ÎÐ~7yc�d{}��|d��{}Ê5»�y½��{}���XÅ�y��CÊAyGÅWÅ�ÍÊ5��� �i�_Ë�� �_���?~`y�{}�
70
�4�"!��'1���b}8M�"* 152
µAcm
���->��4]«��b}�)65*����h��0�!"$�3.�#�_1)@ HH ����� Rinzel ������6#�"*���<�!"1��������"������EDN,�OKJ HH ���W��6#�"*C�n1M�"(X1M�"�4�X<����_$ Rinzel �7�¤��64�"*���!�3d���������U<���!"$�� j��Nª ªÑÈ�0�*
��ÍS~�¼5�?~`y�~5{�Éq~7y�|�É HH x y Rinzel »��B»i¼Ay����Xz�{}|d���%y"y������ �i�_Ë`Ë �_���?~`y�{}�
Ò �¶Ó�ÔKÕ �� ��� �ÖT)!��58a>��������;>�!��)(�*���+��2�7�B14�5����+�0�*C1V�_$�!��->�!�!��)(�*���+��->�!"�76#�"*�1?!�]t3)�5*��h<��"�50�!�Z��� ��64�"*�14��>�!��)(�*���+��o�73d����!�8#*C�u>�!�!��"0�!"$M�����C@)b}!�*W�C�56)!�0X@)�"8#!"151wT)�#�"8w��!Q<�3w�"*�3.�ET5��@�����C��!�bo��b}8M�"*×,-����+`1Ø�f>�!"��6#�"*×��!"$�1)�-!"��3)3 ���"35��@�*Ù!��"�C!�]}�-�C(�!��Ú��b}8M�"*��'14�×����+`1���"�����a���_@�8#*��'1h��*��"*�*9�)�������;>��4]t� ��6#�"*Û�E!�bl�_�)�"!"1a1#85@Ü���9���:�76#�"*;��!�!"$�1)�?��!"1#!�(C$��5(C�)������������_��!"@i+����W>�!��#�C��*W�5���X�CZC*��"*W!����"Zo���?��3)�A@�*��m���"!����"34�'��*W���"!"1#8#!"@