OK, so if I read this correctly, you want one wave to travel across in one direction each time you press a button.
And I gather that the terminal condition is all LEDs OFF.
OK, here is my basic button-and LED code:
// Blink without "delay()" - multi!
const int led1Pin = 13; // LED pin number
const int led2Pin = 10;
const int led3Pin = 11;
const int button1 = 4;
int led1State = LOW; // initialise the LED
int led2State = LOW;
int led3State = LOW;
char bstate1 = 0;
unsigned long count1 = 0; // will store last time LED was updated
unsigned long count2 = 0;
unsigned long count3 = 0;
unsigned long bcount1 = 0; // button debounce timer. Replicate as necessary.
// Have we completed the specified interval since last confirmed event?
// "marker" chooses which counter to check
boolean timeout(unsigned long *marker, unsigned long interval) {
if (millis() - *marker >= interval) {
*marker += interval; // move on ready for next interval
return true;
}
else return false;
}
// Deal with a button read; true if button pressed and debounced is a new event
// Uses reading of button input, debounce store, state store and debounce interval.
boolean butndown(char button, unsigned long *marker, char *butnstate, unsigned long interval) {
switch (*butnstate) { // Odd states if was pressed, >= 2 if debounce in progress
case 0: // Button up so far,
if (button == HIGH) return false; // Nothing happening!
else {
*butnstate = 2; // record that is now pressed
*marker = millis(); // note when was pressed
return false; // and move on
}
case 1: // Button down so far,
if (button == LOW) return false; // Nothing happening!
else {
*butnstate = 3; // record that is now released
*marker = millis(); // note when was released
return false; // and move on
}
case 2: // Button was up, now down.
if (button == HIGH) {
*butnstate = 0; // no, not debounced; revert the state
return false; // False alarm!
}
else {
if (millis() - *marker >= interval) {
*butnstate = 1; // jackpot! update the state
return true; // because we have the desired event!
}
else
return false; // not done yet; just move on
}
case 3: // Button was down, now up.
if (button == LOW) {
*butnstate = 1; // no, not debounced; revert the state
return false; // False alarm!
}
else {
if (millis() - *marker >= interval) {
*butnstate = 0; // Debounced; update the state
return false; // but it is not the event we want
}
else
return false; // not done yet; just move on
}
default: // Error; recover anyway
{
*butnstate = 0;
return false; // Definitely false!
}
}
}
void setup() {
pinMode(led1Pin, OUTPUT);
pinMode(led2Pin, OUTPUT);
pinMode(led3Pin, OUTPUT);
pinMode(button1, INPUT);
digitalWrite(button1,HIGH); // internal pullup all versions
}
void loop() {
// Toggle LED if button debounced
if (butndown(digitalRead(button1), &bcount1, &bstate1, 10UL )) {
if (led1State == LOW) {
led1State = HIGH;
}
else {
led1State = LOW;
}
digitalWrite(led1Pin, led1State);
}
// Act if the latter time (ms) has now passed on this particular counter,
if (timeout(&count2, 300UL )) {
if (led2State == LOW) {
led2State = HIGH;
}
else {
led2State = LOW;
}
digitalWrite(led2Pin, led2State);
}
if (timeout(&count3, 77UL )) {
if (led3State == LOW) {
led3State = HIGH;
}
else {
led3State = LOW;
}
digitalWrite(led3Pin, led3State);
}
}
You may care to adapt it to your circuit and try it out. While it does not do what you presently want, it contains all the core functions necessary to do so, which I will look into later. Sadly - perhaps - I have to go to work just now. 
In the meantime for clarity, can I ask you to go back to your first post, select "modify", selectively highlight the code section and use the little "scroll" icon with the brackets to mark that as code?
